Y is a vowel in Guyana.
The pronunciation of "Guyana" is /ɡʌɪˈanə/ (ODO). Like in the word "guy" or "guard", the "u" is silent. As in "guy" or "by", the "y" is an "eye"/"I" sound /ʌɪ/ or /aɪ/. So it is a vowel sound. (If you regard the "uy" jointly as making the /aɪ/ diphthong, I'd still argue that it should be considered a vowel.)
Where a sound is transcribed with /ɪ/, /aɪ/ etc these are vowels; /j/ would be the consonantal equivalent. (Some linguists transcribe /aj/ but no major dictionary appears to. In any event, most sources define a diphthong as a coming-together of two vowels.)
Addressing the counterargument
Some say they "hear" the "y" in "Guyana" as a consonant. There are two ways of interpreting this statement.
1. Perhaps they pronounce the first syllable with a monophthong, e.g. /ga/, and then they pronounce a second syllable starting /j/. But this isn't the pronunciation offered by dictionaries, which agree that the standard pronunciation rhymes the first syllable with "eye" and "try". In my answer I address the standard pronunciation.
2. A second possible interpretation is that they hear a /j/ after the "eye" diphthong. This can indeed often be heard (though it is not in the dictionary transcriptions), but you hear it too in "bionic", "Siamese", "iota", "triangle". The "y" cannot be making this /j/ sound; the "y" is making the "eye" sound; the /j/ comes in naturally between two vowels, as in the other examples I've cited.
NicholasCastagnola mentioned the variant spelling "Guiana". If the "y" in "Guyana" were considered a consonant, we'd have to consider the "i" in "Guiana" a consonant. And whereas I assume we're all agreed that the "y" in "by" or "buy" is a vowel, if we accept the above counterarguments we'd have to consider the "y" in "buying" and maybe even "trying" a consonant - so the "u" in "buying" and "Guyana" would have to make the diphthong all by itself, unless the "y" is pulling double-duty (as others might claim with respect to "trying").
According to phoneticsontheweb:
Sometimes, the letter y is a consonant, and other times it is a vowel. The rule for telling the two apart is simple: The letter y is a consonant when it is the first letter of a syllable that has more than one letter. If y is anywhere else in the syllable, it is a vowel.
The word "Guyana" has three syllables, and the "y" is part of the first syllable, and not the start of it.
Yougowords cannot be considered a reliable counter-source. It claims that "Guyana" has "two syllables", which is incorrect. It also regards "by" as having "zero vowel". To be fair, there is a tradition that some people follow of restricting the definition of "vowel" to the five letters a, e, i, o, u. But the whole basis of the present discussion is that we are regarding "y" as a letter that sometimes makes a consonant sound ("yet") and sometimes a vowel sound ("by") and discussing which categorisation works better for the "y" in "Guyana".
I am not sure that it makes scientific sense to classify letters (rather than sounds) into vowels or consonants. Presumably, by convention, we would classify the "u" in "guard" as a vowel, but since it's silent, it's making neither a vowel sound nor a consonant sound. It is worth noting that where the vowel/consonant distinction is used in English grammar rules, it is the sound that matters, not the spelling - so "unit" is always preceded by "a", for example, not "an".