WASHINGTON—The American Israel Public Affairs Committee has drawn criticism from several progressive new lawmakers, including Rep. Ilhan Omar, who said in tweets this week that the lobbying group holds financial sway over Congress.

The Minnesota Democrat, who was heavily criticized for her remarks, has apologized for messages many viewed as anti-Semitic because they played into stereotypes about Jews and money. But Ms. Omar said she has no regrets about calling out the influence of the group, known as Aipac.

Aipac...