Buying a bike without riding it is a terrible idea. Similarly terrible ideas would include ordering a mattress without laying on it, laying down cash for a guitar without playing it, and purchasing bike shorts without trying them on. And I agree that any shop that won't let you test ride a bike is pretty suspicious. I'd try to take my business elsewhere.
That said, we live in a non-ideal world. Sometimes the perfect deal comes along, or you just plain need something and have to take a chance and buy it sight-unseen. In this answer, I'm assuming that you've tried to find other ways of riding the bike (asked friends, looked for other shops, etc.) and have failed to get that test ride.
There are ways to make an imperfect frame fit, if the sizing isn't too far off. Extra-long stems and seatposts, barends, saddle rail extenders... all of these are gadgets that you can buy to help get yourself on an ill-fitting bike. If you make some good guesses, yu can avoid using any of these contraptions.
What can you do to mitigate the risks?
First off, I'd read up on bike fit as much as possible. This way, you can make guesses that are informed guesses.
With any frame, even a custom-built one, you'll end up adjusting saddle-to-pedal distance, reach, handlebar height, and frame size. It sounds like you've narrowed things down to a couple of sizes; that's great, as it makes your job much easier.
Narrowing it down
In the two sizes you've chosen, which one will allow you more latitude when dialing in your fit?
If, for example, the smaller frame barely has enough standover height, that's a point in favor of avoiding the larger one. If the smaller frame would need an extraordinarily long stem to get a good reach (that's the saddle-to-handlebar dimension), that's a point in favor of the larger frame.
And so on. Tally up all of these factors and see which frame makes the most sense.
Can this possibly work?
Speaking as someone who owns a built-to-order road bike, who put money down before the thing was even built: It's possible to do this if you know what you want and know what's important to you.
An awful lot can go wrong. You've made a lot of guesses, based on your experience with other bikes, reading spec sheets, and just plain hoping your gut feelings are right. If you can't afford to los the money you'd spend on this bike, I'd think twice before ordering.
If it's possible to wait a while and test-ride a bike then (maybe you're going on vacation later this year and can hit some shops then?) (is a road tide coming to town? Put out a classified ad) I'd do that.
But if not, try to ride some other road bikes and get a good idea what's important to you in road bike fit. Remember that you're gambling a bit with your money, make the best guesses you can, and hope. The worst-case here is that you end up selling the bike and taking a loss.