At what point do I buy inventory for an online business?
Get the website up first, and as long as your margins are good, you can allow for extra delivery time upfront, as long as you let your customers know when they can expect to receive their orders.
If you are concerned about this, you could pre-order the top two or three items you think will sell most and have those on hand, but keep in mind that the whole idea behind many online ventures is to keep a limited amount of stock on hand.
That said, continue looking for a good drop-shipper you can work with. There are many who would be amenable. Just make sure you can
keep your margins where they need to be, and that you are not paying retail for something that should be priced wholesale or better.
All the best.