do you think Dorothy and the Scarecrow will let the Tin Woodman join them?

I suppose a reader could say that Dorothy and the Scarecrow are unlikely to let the Tin Woodman join them; however, I do think that would be a tough point to prove. Yes, I think it is entirely likely that they will let the Tin Woodman join them on their way to see Oz. The best supporting evidence that I can give to this idea is that Dorothy has already proven herself willing to invite fellow travelers with her. She invited the Scarecrow just a bit earlier because he would like to get some brains. Dorothy and the Scarecrow discover the Tin Woodman frozen in place because his joints are rusted up. The Woodman is deeply appreciative when they use oil to get him moving again.

"I might have stood there always if you had not come along," he said; "so you have certainly saved my life."

The Woodman asks the two other characters why they are travelling as they are, and Dorothy explains the situation. The Woodman ponders her answer, and he asks if Oz would be able to provide him with a heart.

The Tin Woodman appeared to think deeply for a moment. Then he said:
"Do you suppose Oz could give me a heart?"

Dorothy and the Scarecrow both agree that Oz should be able to do that, so the Woodman asks if he can join them. I don't believe that readers are really ever meant to question whether or not Dorothy will say yes. She is a loving character who assumes the best about people. She doesn't even hesitate in extending the invite to the Tin Woodman. The Scarecrow has been presented as a jovial and happy character too, and he isn't likely to say no after having been in the same position quite recently.

"Come along," said the Scarecrow heartily, and Dorothy added that she would be pleased to have his company.