Sounds like the place your staying at/want to make a reservation doesn't do on line bookings and they want your credit card info to reserve/pay for it, correct?
I have done the same thing in the old days with hotels in Mex. Both countries are shady, just watch your credit card bill for any goofy charges after giving out the card info or better yet, book the trip, hotels, etc through a travel service website that you can pay on line. Call Amex Travel Services if you use them.
I booked a rental flat in France and had to send them my credit card details via email (no online form). I was a bit wary and made a point of telling them to delete the email immediately after. Of course they said there was nothing to worry about.
A month later my bank called to ask if I had made $5000 in purchases in Spain the previous day. I had not.
Annoying? Yes. But they just expunged the charge from my account and sent me a new card, so not really all that big a deal in the end.