Can I use my HSA money to purchase sperm from a sperm bank?

That depends. 

Expenses are deductible (and therefore eligible for HSA since the rules are the same) when they are to treat, prevent or mitigate a disease or affect the natural function of the body, and the expense is for you, a spouse, or a dependent.

So this now gets into areas of law that are controversial.

A heterosexual married couple who are infertile can use HSA money for fertility treatments.

A gay male couple can use HSA money for a sperm bank if one of the partners is infertile and they need the sperm for a surrogacy procedure, but they can't use HSA money for other surrogacy or fertility expenses (like IVF or the surrogate mother) since being two males is not a "disease" and two males having children is not a natural function of the body. (Recent Supreme Court ruling.)

Following that logic, a lesbian couple, or a single female seeking to purchase sperm for IVF, IUF, or surrogacy, could not use HSA money for the sperm, since the lack of a male partner is not a disease.

(This is not necessarily fair, but it appears to be the state of the tax law at this time.  Any changes would have to come from Congress.)