Short answer: most major airlines and many credit‑card travel portals let you book award (miles/points) tickets online — but availability for some partner airlines or complex routings may still require calling the program.

Who you can usually book awards with online (examples; all have web booking or mobile-app booking):

• Major U.S. carriers: American Airlines (AAdvantage), Delta (SkyMiles), United (MileagePlus), Southwest (Rapid Rewards), Alaska (Mileage Plan). (aa.com)
• Big international loyalty programs: British Airways (Avios/Executive Club), Air Canada / Aeroplan, Singapore KrisFlyer, Lufthansa Miles & More (and many other European/Asian carriers). (britishairways.com)
• Some smaller/full‑service carriers also let you redeem online (Emirates Skywards, Qantas, Japan Airlines/JAL, etc.) — check the airline’s “use miles”/award pages. (Examples above are representative; most carriers document online award booking on their site.)

Credit‑card travel portals (book flights with points or “pay with points” online):

• Chase Ultimate Rewards travel portal, American Express Travel (Pay with Membership Rewards), Capital One Travel, Citi ThankYou Travel — you can search and book flights with points through each issuer’s portal. (Note: portals usually book revenue tickets treated like paid fares, not airline award tickets; they have different rules/benefits.) (nerdwallet.com)

Important caveats and tips:

• Not all partner awards are always bookable online — some programs don’t display every partner seat or restrict certain award types to phone agents (e.g., Miles & More notes exceptions; some partners or “world award” routings may need agent help). If you find availability on a partner site but can’t recreate it in your program’s search, you may need to call. (miles-and-more.com)
• Some programs (e.g., Avianca LifeMiles) let you book online but don’t show every Star Alliance partner seat — people often search on other partners’ sites first, then confirm/book online or by phone. (pointswithq.com)
• Award‑search tools (Point.me, SeatSpy, ExpertFlyer, etc.) can help find availability but typically don’t complete award bookings — they direct you to the airline/program site to book. (wsj.com)

If you want, tell me which airline(s) or which miles/points you have and I’ll list the exact program pages (and any known partner/phone exceptions) so you can book online right away.