Learn Turkish: Onların yıkaması gereken kundak çok kirli. - The swaddle they must wash is very dirty.

This is the standard way to form a relative clause of necessity taught in this lesson. Literally, it translates to "the swaddle which their washing (yıkaması) is necessary (gereken)". The verb takes the verbal noun suffix -mA and a possessive suffix to show who needs to do the action. Then, the active participle -(y)An is added to gerek- to link it to the noun kundak.

Since the pronoun is Onların (their), shouldn't it be yıkamaları instead of yıkaması?

You can absolutely use yıkamaları, but yıkaması is also perfectly correct and very common! In Turkish, when the plural pronoun (onların) is explicitly stated, the plural suffix -lAr on the possessed word is often dropped. So, onların yıkaması and onların yıkamaları mean the exact same thing here.

Why do we use gereken with the -(y)An suffix instead of the -dIk suffix, since "swaddle" is the object being washed?

Even though "swaddle" is the object of "wash", the grammatical subject of the relative clause is the entire action of washing (onların yıkaması - their washing). Because "their washing" is what "is necessary" (the subject of the necessity verb), we use the subject participle -(y)An on the verb gerek- (to be necessary), resulting in gereken.