$F_{ST}$ when considering a multi-allelic locus
Short answer: yes, people have formulated ways to estimate $F_{ST}$ for multiallelic loci, e.g. microsatellites. For a review, see here.
Specifically, Nei could define $F_{ST}$ for multiple alleles as
$F_{ST} = \frac{(H_t - H_s)}{H_t}$,
which is to say the proportion of total heterozygosity that is across rather than within populations. This is agnostic to the number of alleles, as it just uses heterozygosity. I believe there are other formulations as well.
There exist alternate statistics such as $G_{ST}$ and $R_{ST}$ (links under each) which are devised specifically for the multiallelic problem, see also the review for further discussion. Basically, microsatellites have high mutation rates such that they deflate the statistic; this will likely be the case for most multiallelic loci.
It's not clear though which statistic works best in practice as far as I can tell.