Să is a conjuction and it's determining the conjunctive mode in Romanian. As far as I know, in Romanian, this situation is different than in other Romance languages. While other Romance languages use infinitive for such situations, Romanian uses the Slavic conjunctive mode. However, it is possible to use infinitive, too.
Would you like to have...? Ai dori să ai... ? (conjunctive)
Would you like to have...? Doreşti a avea... ? (infinitive)
Indeed "să" is a conjunction used to form the subjunctive (that's why it's called "conjunctiv" in Romanian). In everyday language it's more likely to find the subjunctive after verbs like "a vrea"(to want), a intenţiona (to intend), a se gândi (to think of), a putea (can / to be able to) and the like, somewhat similar to French.
"Can I ask for your hand" - "Pot să-ţi cer mâna (în căsătorie)?"