Journalists must do this all the time.
No. They don't.
Does anyone know how this data can be researched?
You'll have to go to the clerk's office at the superior court (or maybe the district court) in Bellingham. Ask the clerk if there is any way to search case files by case type. If not, then you'll have to go through every single file for whatever time period. If there is a way to search by case type, then this probably won't be terribly difficult.
I am trying to figure out what percentage of cases of domestic violence have been decided in the favor of the woman in my particular court in Whatcom County, Washington. I have a suspicion that the court is gender biased and therefore violates the equal protection clause and I would like to do something about it, perhaps a class action lawsuit or somehow involve the Supreme Court.
Ok...for starters, are you talking about criminal prosecutions? The subject header of your post mentions "protection order cases," but that's a different thing from "cases of domestic violence" (although they may well go hand in hand). Criminal cases are not prosecuted for or decided "in favor of" the victim.
Moreover, a mere "percentage of cases . . . [that] have been decided in . . . favor of . . . [women]" is, by itself, meaningless. I suspect that the overwhelming majority of DV/protective order cases are brought by women, so it stands to reason that a majority will be determined in favor of women. Even if it's 75% or 80% that's not an indication of bias. One would need to critically evaluate each case on its merits to determine whether the result was not supported by the evidence.
What you will find out is that, when a person seeks a restraining order, the initial order is often issued based on very little evidence and without involvement from the person against whom the order is sought. The court will then set a further hearing a couple/few weeks later to determine if the temporary order should be extended or made permanent. The person against whom the order is sought must be given notice and will have an opportunity to contest the order. You would probably find that upwards of 90% of applications for temporary restraining orders are approved (regardless of gender). However, the percentage of orders that are extended beyond the initial period will be much lower.
Finally, you didn't mention why this concerns you, but in order for you to take any action, you'd have to have standing (i.e., some tangible, personal interest that would be affected by the outcome of the action). Most likely, this is the sort of thing that would be addressed by the state attorney general.