I agree with Ciara and Katie. Definitely make sure you and the owner are on the same page and everything is determined before you sign the paperwork. Other then that, I don't know if the owner already has tack you can use but I would not but anything too significant like a saddle until you know that you'll be leasing this horse on the long run or something. But maybe get brushes, bathing supplies and just little things like that.
Plan ahead of time to make sure you and the owner are on the same page in terms of the contract and who is responsible for what, make sure you both have signed copies of the lease agreement, make sure the lease agreement clearly states that you are not responsible in any way for pre-existing conditions (whether you know about them ahead of time or not), know exactly what is coming with the horse and what you'll need to get and consider insuring the horse.
Just make sure you it clear with the owner who is paying for what like farrier feed vet.
Also check when the horse was last vaccinated/wormed/vet/dentist etc
I lease and I use the owners tack!