After further digging it appears that, unfortunately, the paper doesn't correct for multiple comparisons after all.
The study used the chi-squared test to compare the frequency of a SNP in patients vs controls, and helpfully it reports the chi-square statistic as well as the p value. You can calculate the p value directly from the this chi-square statistics. For example, for the first SNP in table 1, the Chi-squared statistic (using the appropriate one degree of freedom) gives an uncorrected p value of 0.003. That's exactly the p value reported in the same table, indicated the p value is uncorrected. Which suggests that false positives are likely to be an issue.
That said, it's not clear what the most appropriate correction would be. Although they tested 233 SNPs, that doesn't count as 233 independent tests due to linkage (two SNPs close together on a chromosome are often inhertited together, so if one SNP is significant in the test, its partner will almost certainly be too: effectively this is one test, not two).
I'm not convinced that significance testing makes sense here anyway unless there is more evidence that the control set is a good representative sample of the overall population. One very big danger is that they end up with control volunteers who have family members with ME which could skew results.