Although Richard Carpenter is an accomplished songwriter the majority of the 'hits' and popular songs recorded by The Carpenters were written by other people.
Mixture of both, as kempie attests. Most of their best-known tracks are covers, but a few were co-written by Richard Carpenter and John Bettis. These include "Top of the World", "Yesterday Once More", "Goodbye to Love" and "Only Yesterday". On their original albums, it was roughly a fifty-fifty split between originals and cover versions.
Paul Williams wrote several of the Carpenters songs but I would have to dig out my old LPs to list them. He wrote Rainy Days and Mondays I'm pretty sure.
topkapi is correct. PW also wrote "I Won't Last a Day Without You" and "We've Only Just Begun" (originally used in an advert for Crocker Bank) among others.