I'm not sure how much you have learned about Japanese grammar but I think it would be helpful to breakdown your phrasing a bit just so you know what you're saying. The "sain" is a loan word from English and it's what they use when signing a document. The "shitemoratte" means someone doing something for you. The "ii desu ka" means "is it alright?"
There are many ways to ask someone to do something for you in Japanese but if you don't feel like memorizing many phrases what you have works just fine. You just don't have an object in your sentence so people probably don't know that you're referring to the shirt. You could say:
Kono shyatsu ni sain shitemoratte ii desu ka.
Could you sign this shirt for me?
As for the phrase you wanted to say, you could say:
(Nihon de tabi shinagara ironna hito ni kono shatsu ni sain shite
kureru you ni onegai shiteimasu. Sain shite moratte ii desu ka?)
As I'm traveling Japan, I'm asking people to sign this shirt. Will you sign it, too?
It's a little long, but it gets the message across.