Short answer — most major airline loyalty-program websites let you search for and complete award bookings online, but availability for partner-operated flights varies (some partners only book by phone). Useful airline sites and a few third‑party award search/booking tools:

Airline/program websites that let you book awards online

• U.S. carriers: American AAdvantage (AA.com), Delta SkyMiles (Delta.com), United MileagePlus (United.com), Alaska Mileage Plan (AlaskaAir.com), JetBlue TrueBlue (JetBlue.com), Southwest Rapid Rewards (Southwest.com), HawaiianMiles (HawaiianAirlines.com). (millionmilesecrets.com)
• Canadian / international programs that let you book online: Air Canada Aeroplan, ANA Mileage Club, British Airways Avios, Air France–KLM Flying Blue, Singapore Airlines KrisFlyer, EVA Air, Lufthansa/Miles & More, Avianca LifeMiles, Virgin Atlantic Flying Club, etc. (most of these let at least some partner awards be booked online). (miles-and-more.com)

Important caveat — partner awards and complete itineraries:

• Even when a program supports online award bookings, many programs do not display or permit booking of all partner-carrier award space online. If the website won’t show or let you finish a partner itinerary, you often must call the loyalty program to ticket the award. Examples and community reporting back this up are common. (viewfromthewing.com)

Third‑party tools and paid services that search and sometimes help book awards

• Point.Me — powerful award search across many programs and shows how to book (can guide transfers or booking steps). (onemileatatime.com)
• Roame, SeatSpy, AwardHacker and other award‑search sites — good for finding availability but usually you still book on the airline/program site or by phone. (roame.travel)
• Specialist paid services / brokers (e.g., AwardBird and other award-booking agents) will search and can book awards for a fee — useful when the airline site won’t book a partner itinerary. (awardbird.com)

Practical tips

• Start on the program whose miles you plan to use — that’s the site that can actually ticket the award. If you need partner space that the site doesn’t show, call the program. (viewfromthewing.com)
• If you plan to transfer transferable points (Chase, Amex, Capital One, Citi, Bilt), use a search tool (Point.Me, Roame, SeatSpy) to confirm bookable space on the exact program you’ll transfer into before moving points. (onemileatatime.com)

If you want, I can:

• List direct award‑booking pages/links for specific programs you care about (U.S. programs, Aeroplan, KrisFlyer, Avios, etc.), or
• Help search availability for a particular itinerary and show which programs/websites let you finish the booking online.