I am writing an email in Italian to a female HR manager. I was going to write "Buongiorno Sig.ra ____", but I don't know if this would be proper given that she may not be married. What is proper in this situation?
Probably neither. Since in Italian whoever as a laurea (the lowest universitary degree) is entitled to be called dottore, or dottoressa for a female, chances are that whoever holds a high-level post in a firm or other organisation has this title. So to be on the safe, you may use dott. for a male and dott.ssa (or dott.) for a female.
As for signora / signorina, the use of the latter is dwindling, and the trend is towards a generalised use of signora, whatever the marital status of the woman involved. For more on this, if your Italian is good enough, see, on the website of Accademia della Crusca, what linguist Paolo D'Achille explains.
In my experience, the most common solution is avoiding the problem by using a different salutation (for instance, just buongiorno) or switching to a first-name basis earlier than normal practice would suggest (in particular, gentile Anna would sound perfectly OK between coworkers in a firm with a relaxed culture).
I don't tend to 'doctor' people I don't know, and I very seldom look their title up. But that might just be a cultural preference; I can see other people doing differently, in contexts where degrees have a higher perceived importance.
You usually start formal letters with "gentile" (literally kind), which can be followed by the recipient's "titolo" (title, qualification). If someone is in a public position they will usually have some sort of profile online that tells you what their qualifications are. Check that and write "dottoressa" (f. graduate) if she has a degree, or you can also write "professoressa" (f. professor) if she teaches. You have to check though, if she doesn't actually have a degree it would be awkward to call her dottoressa.
Even if she has no qualification I would avoid "signora" or "signorina" and simply go for "gentile" + her name and surname, in this order. "Signora" and "signorina" feel dated and there's always a chance you make her feel old or weird if you use the wrong one. If your tone is right your letter can sound formal and polite enough even without "signora"/"signorina".