i do an offsite lease, therefore, i own all of my own stuff as the owners did not provide anything except for a cheap halter (not even a lead!).
Ask the horses owner or whoever you are leasing them from first to see if they already have tack fit for the horse. (Usually most lease horses come with tack or at least and saddle and bridle). If not I would talk to the owner about what the horse currently uses (martingale, breast collar, polos or boots etc.) to fit the horses needs. Then I would search the tack items you need - I would recommend Dover for quality and great looking tack!!
If you don't have a tack store where you live ( like me I live in Serbia where it's impossible to find anything) then I recommend ordering it from DIVOZA they ship nearly to every county. I hope you will like it.
It really depends on the horse's owner. Some like to lease the horse with all his equipment and others don't.
The best thing ro do in your case is to ask directly the owner of the horse you're leasing so you'll be aware.
Good luck :)