Finding the cause of under-extrusion is very hard as a lot of parameters of the print process can influence this. There are some nice websites that describe these problems in detail. From your question it is unclear what you have done to solve the problem, or if you have printed products after the vase mode and shown us a picture of that (this eliminates a lot of possible problems).
A nice overview is given by Ultimaker, but other sources may help you to find the root cause, e.g. Simplify3D. If the issue is related to the filament and hot-end, Printrbot, Trideus and Rigid.ink may help you solve the problem.
Important is to isolate your problem! Not knowing what printer you have, your printer has (or potentially has) the folowing modules/elements that may be causing the underextrusion:
Note that to find the root cause you should tackle this by elimination, this way you make sure that certain modules are not causing the problem. Also keep in mind that the vase mode prints a single outline/perimeter shell and won't make any retracts (so the Z axis will continuously rise), in which defects are shown instantly. Please, take a close look at your normal multi perimeter print.
Under extrusion is probably one of the hardest to find the direct cause as there are so many variables to consider. Please find below some of the variables that can affect your printing quality marked in bold face.
Material and material settings
The material you use needs to be resembled correctly, so it is important and easiest thing to check first if your print is suffering from under-extrusion due to incorrect material settings. The material settings in your slicer (or the material profile on your printer for the more fancy printers) should match the material you are printing. So please check the filament diameter with a caliper and measure the diameter at various points; take the mean diameter of at least 3 to 5 measurements. Furthermore, temperature is also an important factor; too low temperatures will cause that the extruder has to push harder as the material is less viscous due to the fact it is not heated properly. Note that this can also happen if the flow of the filament is too high and the heater cannot keep up. It is these high pressures that cause the under extrusion as it may not flow fluidly. In contrast to too low temperatures, too high temperatures, can also cause problems. Very high temperatures can change the structure of the material, this is often referred to as carbonization causing deposits (clogs) in the nozzle. A word of advice, Please check your filament spool/box (or sometimes a paper in the box or bag) for the proper temperatures.
Next to the temperature, other important material settings are the print speed, the layer height and the nozzle size as these properties further define the rate at which the filament volume is deposited. For instance, a too high of a volume flow not only can lead to the previously mentioned cooling of the nozzle, but also is limited by the diameter of the nozzle, you just cannot push more through the nozzle is capable of as the friction will increase (the smallest opening in the system determines the maximum rate of volume flow). If you do, this will lead to under-extrusion. To find the optimum between speed and temperature, a good balance between these needs to be found. A typical way to do that is by the use of printing calibration temperature towers, preferably at various speeds. To print faster, you need a higher temperature, but printing at lower temperatures because of overhangs, you might need to decrease the speed to get a proper extrusion (and maybe also part cooling).
Don't just focus on the hot end part, also take a closer look at the filament spool itself, or better, how the spool unrolls. Is the spool of filament unrolling correctly/freely without a lot of friction (does it make sharp bends, or does it go through a tube having friction from its container to the extruder), or is the filament not correctly wound causing tangled filament (which create a lot of friction preventing enough material to be transported to the hot end) which could stress the extruder.
For some materials that are hydrophilic (they attract water and trap it in the filament, this happens e.g. with PLA, PVA, Nylon and maybe even more) printing the filament with contained water, the water will turn into steam causing bubbles in the deposited filament and interfere with the flow deposition. This effect sometimes makes a distinct sound like popping bubbles. Always store your filament in a sealed container or bag and use desiccants bags. Moisture can cause damage to the printer as the filament swells when taking up moisture; this could lead to various jams. Last but not least, filament with moisture in it has less mechanical properties after printing than dry filament (up to 33% less).
The extruder/feeder and Bowden tube
The extruder/feeder pushes or feeds material into the hot end, or into a tube (called Bowden tube). Under-extrusion caused by the extruder is typically characterized by the fact that filament is not properly fed to the hot end as a result of too much friction in the tube or hot end, too less grip on the filament or filament grinding (the extruder gear 'eating away' the filament). Too much friction could even cause your stepper to tick or click, basically turning back as the pressure on the filament exerts so much pressure that the stepper is rotated back; increasing the feeder tension on the filament (by adjusting the screw on the extruder/feeder would fix that). Grinding is easily spotted when removing the filament; it will clearly show that the gear has worn away circle shapes. Furthermore, filament taken out of the printer should show visible marks on the material as imprints of the extruder gear, if completely smooth, the feeder tension is too less. On the other hand, too much tension on the feeder could flatten the filament, which leads the previously mentioned grinding effect. If you encounter grinding, please assure that you clean the extruder by removing the filament powder and chunks the grinding produced and recheck the extruder/feeder tension before continuing printing again. Be sure that the grinding particles have not entered the Bowden tube as it causes friction. Cleaning them regularly or replacing them once a year is advisable depending on the usage (or once every x kilometers of filament). Furthermore, larger diameter filament (2.85 or 3 mm) can cause additional friction (in the Bowden tube or the extruder/feeder) as towards the end of the spool, the filament is wound tight along a small diameter spool center causing strongly bend filament that exerts pressure as it acts like a spring creating friction at the walls of the tubes.
The hot end can also be a culprit for under-extrusion. Partial blockage of the nozzle as a result of carbonization (buildup of carbon or carbonized material in the nozzle). Even left over material from previous prints inside the nozzle (unflushed residue) may change the volume of the nozzle when the material you printed before needed a higher temperature than the current you're printing. Also try to get good quality filament, it might be that the quality is just not constant for the whole spool. Too clean the inside of the nozzle, a few techniques exist to remove blockage. By performing a "cold pull" or using the
atomic method. Both techniques rely on the mechanism to insert the (cleaning) filament when it's hot and remove it quickly at a lower temperature. E.g. see here or here.
You are printing walls of 0.3 mm with a 0.4 mm nozzle (25% contraction). This might fall in the category of insufficient volume flow. The wall width is less than the nozzle diameter, so it needs to contract. Either way, the volume is too less and may be dragged along until it is flattened. You should increase the wall width, maybe even a little more than 0.4 mm (although that might not be necessary, see edit below).
To support this, the next table shows the volume calculation for the filament (based on extruded cilinder shape), the first row is the volume you need to fill the 0.2 mm layer height (0.025 cubic mm), the second is what you extrude with a 0.3 mm width for a 0.2 mm layer (0.014 cubic mm). From this latter value you can calculate the layer height for a 0.4 mm diameter nozzle to be about 0.11 mm, clearly not filling the gap.
Edit: To further support this, please read this test.
The extrusion width versus strength show the optimum strength when the nozzle extrusion width equals the nozzle size.