The park has been closed due to the corna virus problem but the Wyoming part of the National Park will open May 18th. The fishing season opens Saturday, May 23rd. This article from the park’s website explains this in detail. Be sure to see the map at the bottom of the page: https://www.nps.gov/yell/learn/news/20015.htm
Streams Outside Yellowstone National Park: These still have open seasons in many sections of the rivers.
Madison River (from Hebgen Lake to Ennis) – Weekly Updated Fishing Report
Yellowstone River (Outside the Park) – Weekly Updated Fishing Report
Gallatin River (Outside the Park) – Weekly Updated Fishing Report
Shoshone River, Wyoming – Weekly Updated Fishing Report
Snake River near Jackson, Wyoming – Weekly Updated Fishing Report
South Fork of the Snake River, Wyoming – Weekly Updated Fishing Report
Ruby River, Montana – Weekly Updated Fishing Report
Henry’s Fork Snake River, Idaho – Weekly Updated Fishing Report