How do you commonly syllabicate words that end with -ism? When I pronounce -ism, it sounds like two syllables, (feudalism sounds like 4 syllables, racism sounds like 3 syllables), but in general, a syllable requires a vowel, which we don't have in this case. So technically speaking, does the -ism suffix have 1 syllable or 2 syllables?
The suffix "-ism" is generally analyzed as having two syllables. However, the second syllable is not particularly prominent, and no contrast in syllabifiation is possible for this word in any variety of English that I know of, which may explain why some speakers don't have particularly strong intuitions about this. Also, depending on what linguistic theory you are working with, syllabification might not be considered a particularly fundamental feature of pronunciation, and the syllabic pronunciation of the "m" in this context might be considered to be just a contextual variant of the same sound found in non-syllabic form at the start of meet or the end of team.
Most varieties of English are thought to have sounds called "syllabic resonants": sounds like m, n, l, and in some varieties of the language r can occur not only in the onset or coda of a syllable, but as the nucleus of a syllable. I won't discuss l and r any further here because they are a bit more complicated to analyze.
Syllabic n and m only occur in unstressed syllables, and for many (I think most) speakers there is no noticeable phonemic contrast between syllabic n and m (written in IPA as n̩ and m̩ respectively) and the sequences /ən/ and /əm/. There is certainly no stable phonemic contrast between these sequences that applies to the general community of English speakers.
One thing to keep in mind is that some speakers produce and/or perceive a non-phonemic distinction between [ən], [əm] and [n̩], [m̩]; for these speakers, probably only the latter would be described as a "syllabic resonant". The exact distribution of phonetic syllabic resonants vs. phonetic [ə] + resonant sequences seems to vary somewhat between dialects. For example, this blog post by John Wells indicates that in "RP-style English", [ən] is used instead of [n̩] after a nasal or vowel (Wells gives the examples of common and lion). For more information about phonetic details like this, see the last section of this answer.
There is little contrast between syllabic /əm~m̩/ and non-syllabic /m/
There are fairly few contexts where syllabic resonants have a clear phonemic contrast with the equivalent non-syllabic resonants. As far as I can tell, the only contexts where there is a strong phonemic contrast between e.g. /ən~n̩/ and /n/ are after a vowel, as in "Owen" vs. "own", after a liquid, as in the near-minimal pair of "Aaron" vs. "cairn" (for an American English speaker) or "film" vs. "vexillum", and (to a lesser extent) before a vowel, especially a stressed or unreduced vowel, as in transnational vs. fascination.
Since the "m" in -ism is neither preceded nor followed by a vowel, there is no possible phonemic contrast between syllabic and non-syllabic pronunciations. Most speakers think it sounds syllabic (i.e. /əm/), and this fits in better with the usual structure of English syllables, but some people apparently don't hear this. (There is not a huge difference between the sound of n̩ and n, or m̩ and m.)
In poetry, it is often possible to use "compressed" pronunciations where a syllabic resonant doesn't count as a syllable: e.g. words like heaven and given may be treated as monosyllables (these may be written as "heav'n" and "giv'n").
Syllabification isn't based on the presence of a vowel letter in spelling
Syllabic "m" does occur in some contexts where it is written with a preceding vowel letter, such as (for many speakers) in the words bottom, fathom and bosom. However, unfortunately, I don't know of any commonly used words where the sequence /ɪzm̩/~/ɪzəm/ is written with a vowel letter before the letter "m", so it doesn't seem to be possible to do a rhyme test. However, tchrist pointed out the similar word "rhythm" which I think most speakers would feel has two syllables.
I wouldn't put much weight on the spelling as a source of information about how to pronounce the word, though—many words with syllabic resonants are spelled without vowel letters before the consonant letter, like fiddle and centre, or brand names like Tumblr and Flickr (which don't even have a final silent "e").
More phonetic details
Throughout this answer, I've been focusing on the phonological contrasts at the "surface" level. The phonetics of syllabic resonants vs. schwa-resonant sequences, and the possible implications for the "deeper" levels of phonology, are quite complicated (see e.g. The Phonetics and Phonology of some Syllabic Consonants in Southern British English, Zoë Toft, 2002) and I haven't studied them enough to be able to say anything helpful in this regard.