Learn Icelandic: Hvað sem er getur gerst á meðan við flökkum.

Only the first word (hvað) declines based on its role in the sentence, while sem er stays exactly the same. Because hvað is a neuter pronoun, its nominative and accusative forms are identical. If it were the object of a verb requiring the dative case, it would change to hverju sem er. In this sentence, it is the subject, so it remains in the nominative.

Since hvað sem er encompasses 'anything', could I use a plural verb instead of the singular getur?

No, you must use a singular verb. In Icelandic, hvað sem er is grammatically treated as a neuter singular pronoun, regardless of how many theoretical things it refers to. Therefore, it always takes a third-person singular verb like getur.

Doesn't the modal verb geta require the following verb to be in the supine (sagnbót)? Why doesn't gerst end in an '-i' or '-t' like most supine verbs?

gerst actually is the supine form! The verb here is the middle voice gerast (to happen). When forming the supine of middle voice verbs, the -st suffix is added to the active supine. The active supine of gera is gert, and when you add -st, the 't' is dropped for ease of pronunciation, leaving us with gerst.

Can I drop the á and just say meðan við flökkum?