We often use the word "play" with "with" e.g. he played with his toys / his friends. We are saying what we are using when playing, or who we are with. You play with someone or something. In this sentence, Tom wanted to "play with" someone, but this was not possible. So he could not play with anyone. The writer has used the construction "to have + something/somebody + to-infinitive". e.g. I am busy - I have some work to do. In English, it is acceptable to finish sentences with a preposition. We do it when we are using prepositional verbs (like "play with") or phrasal verbs. Dannboy's variation is also good, though formal and less common.

Technically the correct sentence is, "Tom had no one with whom to play." However, this is very, very formal, and almost no one speaks this way in everyday conversation, or even in a lot of more formal writing. Americans, at least, actually tend to dislike when people speak this way.