Hotel points and airline miles are different on this.

Hotel points are generally only earned when booking through the hotel program website or toll-free number. In some cases corporate booking sites may be honored (but in some cases of those only for high elite status hotel program members!), but hardly ever third-party booking sites. The only way to make use of third-party booking sites while still earning points is to use them to leverage a particular hotel program's "best rate guarantee", but how well that works and how easy it is and exactly how to do it varies a lot from program to program. (It's generally worth pursuing only if you are finiding the third-party rates to be signficantly lower and yet for the same terms, like cancellation period, etc. And in no case can it be "blind" booking sites or rates that are "last one at this price" or "only valid for the next x hours" kinds of rates.) Btw, in many hotel programs, hotel-specific discount rates that you arrange with the hotel itself may not earn points either!

Airline miles are quite different. As long as you're booking a fare code (the single letter like Q, K, L, etc, but these vary by airline and sometimes even route as to their meaning and which ones count and which don't) that's valid for earning miels, it generally doesn't matter where you buy it. (But niether does the price change where you buy it, so why not buy it at the airline site, unless you're putting together an itenerary that includes different airlines that don't partner with each other.)