Originally Posted by
ty97
I've never encountered a party or third party site where could not enter your FF number so I do not anticipate any issues there. If you ever did encounter it, you could request missing mileage credit after the fact, but I really don't think you'll encounter this issue
Agreed. Although if it turns out to be necessary there are a number of ways to accomplish it beforehand, the easiest of which is to pull up the reservation via confirmation number at the airline's web site and enter your FF number there, or call them, or ask at the counter if checking a bag or the gate agent, etc.
Google Flights is good for searching a broad range of dates, and you can use the airline dropdown to filter by alliance.