Where are you doing your searches? This is a ski-area. Lots of pensions are open.
Here is a link to the city of Gimmelwald, list of accomodations there: www.gimmelwald.ch/en/accommodation/stay.htm
Now, the Pension Gimmelwald is closed from the middle of April until the 1st of June -- ski season is over, and the staff takes their vacations. It is possible you are encountering the same thing at other hotels -- but even if their website says they are closed, send a e-mail anyway. Depending on how their booking dollars were for ski season they may not take as long of a hiatus.
Leisis heart Place is a smaller pension than some -- and as such are flexible about dates (and rates). Contact them directly, don't beleive second-hand info on them.
Here is a link to the Sunny Days B&B at the bottom in Interlaken: www.sunnydays.ch/. They are open year-around, only close for their vacations.
An option half between Interlaken and Gimmelwald is Murren -- www.muerren.ch. Go to Accomodations, they have more hotel options, is a bigger city, and not quite as far up the mountain.
Among the Murren options, I personally like http://www.eigerguesthouse.com/. (Maybe closed 4/11-5/21, double check.)
Others prevoiusly recommended on this board:
http://www.alpenruh-muerren.ch/en/ and http://www.hoteleiger.com/.
No matter what 'closed dates' are on the website -- contact the hotel -- they are as bad (worse) about updates websites as Americans!
(Please let me know if I've got a dead link.)