I am in this boat right now, purchasing a horse that we will pick on in early October. I have been working with the BO for several weeks to sort all this out. I'm not worried about vet and farrier stuff yet (the sellers have conveniently just wormed and shod the horse so we're good for 5-6 weeks - make sure that when you finalize your horse purchase, you ask when he is due for all those things), but we will likely start off with the ones used by the BO unless we need something more specific. I've also told him that we will need a stall for October and that we are doing a two week trial. He's assured me he has a stall for us. To be fair to him, I've offered to pay for the whole month regardless of whether or not we keep the horse. Should we return the horse after two weeks, he would have two more weeks to find a new boarder. I think this act of good faith goes a long way to sending a message to the BO that you understand he/she is running a business.
Other questions you should ask your boarder are about the extent of the care. For example, my horse has a pink nose and requires sunscreen so I checked to make sure they were willing to apply it (otherwise I'd have to stop in every morning which would be difficult). What is included, what is not? I asked about borrowing tack until we find our own (he's fine with it). You should ask about scheduling arena time if there are a lot of other boarders and/or lessons. Also, does the BO require a long-term commitment like a lease? Or is it month-to-month? It's important to know these things ahead of time so no one is left feeling frustrated.
Finally, if the weather is cooperative and depending on the horse, you may be able to keep your horse outdoors for a couple of weeks while the BO finds a stall for you. Not ideal, but for some horses that are out 24/7, perfectly adequate. Good luck! And like everyone else has said, talk to your BO. They should be open and willing to communicate with you on all these issues. If they are not, it does not bode well for your future relationship. Mine is a great resource and I can call him anytime to ask questions about anything.