No harm to ask

Without knowing exactly, when and how long your vocation is, what you promised them and what is the condition of part time, I would say that you still have room to negotiate. You did not lie about you can start immediately. You just have a vacation planned. Tell them the truth once you get the written offer. Show them the flexibility and you are willing to accomadate. If you have to cancel the vacation, let the manager know that you did so as you really love the job at least.