Camping is available at the Mt Field National Park camp ground at the base of the mountain:
https://parks.tas.gov.au/explore-our-parks/mount-field-national-park/mount-field-camping
Hut accommodation is available at The Government Huts near Lake Dobson at 1,000m elevation: https://parks.tas.gov.au/explore-our-parks/mount-field-national-park/mount-field-government-huts