In this case the subject is hidden, so you could say "how does one spell your/his/her last name". The best way if you don't want to say "you" is "how is your/his/her last name spelled".
Not sure why, but when we say "o seu sobrenome" it refers mostly to "your last name" and when we say "seu sobrenome" (without article) it may refer to "his/her last name".
Anyway, "seu" as "his/hers" is not usually used when talking to someone face-to-face because of the ambiguity.
You can always omit the article before a possessive, that actually happens a lot, especially at the beginning of sentences. Even in this case, if you say "Como se soletra seu sobrenome?", there's a 99% chance that you're asking "How do you spell your last name?", unless you mentioned someone else in the conversation, e.g. "Sabe a Ana? Como se soletra seu sobrenome?" ("You know Ana? How do you spell her last name?").
It's mostly a matter of preference based on regional / dialect differences.