Are there any other English syllables without vowels, besides "thm"?

Solution 1:

Occasionally -sm does the same thing: chasm, schism, etc. As I pronounce them, these are all two-syllable words.

Having said that, I would question your premise that "all English syllables have a vowel sound". There are in fact a great many English syllables which don't have any vowel sound at all (in most US English dialects, as discussed below), but rather have a syllabic consonant:

button

tanner

pickle

The second syllable of all of these words, though spelled with a vowel, is typically pronounced with no vowel sound at all between the medial consonant and the final consonant. Instead, the final consonant is elongated into a syllable of its own. In pickle, for example, there is no vowel, not even a schwa, between the [k] and the [l]. As soon as the [k] is released the lateral contact on the [l] begins, and the [l] sound is drawn out for the full length of an unstressed syllable. In my dialect, at least, all words ending with an unstressed syllable containing [n], [r], or [l] are pronounced this way.

Different dialects handle this differently, however. In British English, for example, tanner often has a final shwa and no [r] sound at all, and the handling of unstressed final [n] as in button varies quite a bit even within North America.

Solution 2:

I don’t know of any satisfying answer aside from JSBձոգչ’s -sm as in chasm. But, here are some probably unsatisfying ones with resononants that are (possibly) syllabic but don't have any written vowel letters corresponding to their syllable.

Words with an ambiguous number of syllables

This depends a lot on your accent, but for many people at least some of the following words are disyllabic, with the second syllable composed only of syllabic “l”: snarl, oil, owl, pearl. (For me, the first two seem a lot like they have two syllables, while the last two seem more like monosyllables or at most sesquisyllables.)

Likewise, with possibly syllabic “r”, we have choir, coir, hour.

But speakers’ intuitions about these words vary wildly: see "Why the extra syllable in words like these ending in -r and -l?" and "How many syllables are in the word 'hour'?", as well as the paper "Sesquisyllables of English: The structure of vowel-liquid syllables" by Lisa M. Lavoie and Abigail C. Cohn).

Relatively well-established loanwords that use non-English spelling conventions

syllabic “l”: dirndl (from a Germanic language), axolotl and various other loanwords from Nahuatl with -tl, as going mentioned

Solution 3:

How about those containing a y?

For example:

• party
• many
• patchy
• syzygy
• xylophone

They have a vowel sound, but are not "spelled accordingly"

There are also some obscure words like axolotl and dirndl. From wikipedia there is also crwth and cwm.