Learn French: Nous avons vérifié la disponibilité de l'hébergement. - We checked the availability of the accommodation.

Because the sentence uses de (meaning 'of', as in de l'hébergement), you have to combine the preposition de with lequel. Since hébergement is a masculine noun, de and lequel contract to form duquel. You could ask, Duquel avez-vous vérifié la disponibilité ? (Of which one did you check the availability?).

I always get tripped up by 'accommodation' in English because it's uncountable. Is hébergement uncountable in French?

No, un hébergement is a standard countable noun in French. You can easily say des hébergements if you are talking about multiple places to stay or different lodging options, whereas in British or international English 'accommodations' can sometimes sound unnatural.

Could I use contrôler instead of vérifier for 'to check' here?

It is much better to stick with vérifier. Vérifier means to ascertain that something is true, correct, or in a certain state (like checking a calendar or availability). Contrôler is often used more for inspecting something against rules (like checking train tickets or passports) or having authority over something.

I noticed disponibilité ends in -ité. Are these types of words always feminine?