Yes, I think there's a good chance that Laura Ingraham will become Trump's press secretary. She's a Trump loyalist. She's good at being a mouthpiece for the right. She'll probably be good at stonewalling at press conferences (if the Trump white house gives any). So what's for Trump not to like? I won't be watching those press conferences, but I bet she'll be there.
Trump has given Laura Ingrahm a great deal of support and in turn, she has supported him in turn. He is likely to have a press secretary that is very favorable to him and will not question what he says or does. She will not question anything that he says and this is what he wants.
Laura Ingraham has expressed her desire to be elected Trump's press secretary, but she doesn't stand a chance. She is not nearly as extreme or conservative as Trump's many supporters are. Plus, it's clear that Trump is sexist enough to not consider any women for his cabinet, so Laura Ingraham won't be elected to that position.
Laura Ingraham has put too many personal opinions out there in the world to be selected as Trump's press secretary. Trump needs a person that doesn't have much history, that isn't going to come across as in favor of one particular party or ideology. Unfortunately that is not Laura Ingraham.