There is no lodging, other than campgrounds, inside Joshua Tree National Park. As a result, you'll have to look to the inns and motels in the towns that surround the park.
You can find rooms in the town of Joshua Tree; in Twentynine Palms, where the park's main visitor center is based; in Indio; Desert Hot Springs; the Yucca Valley; and, if you don't mind an hour drive to the park, in Palm Springs.
Desert Hot Springs Chamber of Commerce
Joshua Tree Chamber of Commerce
Palm Springs Chamber of Commerce
Yucca Valley Chamber of Commerce