I just wanted to note that, in my experience, orgs. get money by forming connections with the people giving out the money and leveraging them. I'd like to take our situation as an example - I'm not sure how applicable this will be to your own situations, but I hope it will be to some useful extent.
At Illinois, we've got a board of several students (none of whom I think we know this time) and a dean (whom we know pretty well and have traded some favors with) that hands out the vast majority of our cash. I imagine there are similar structures at most other institutions. This structure basically divides the organizations into three groups.
The best-funded organizations represent enough members or are constituents of a large enough voting bloc (e.g. frats/sororities, larger ethnic groups, entire colleges of the university, etc.) that they have a member of their own or of an allied organization on the board. Those are usually the groups getting tens of thousands of dollars. As a small organization by Illinois standards and one not representing any strong identity, we're not in this class.
The next class are groups that have some connections on the board. We're in this class - we know the dean well and sometimes know one or more of the student board members. For this and other reasons, most of our funding proposals fly at this point, so we get several thousand dollars each year at this point, which is great. I think most teams would be quite pleased with this.
Finally, there's everyone else who get nothing or effectively nothing. Until a few years ago, when we forged these connections, that's where we were and it sounds like that's a where a lot of you are, too.
So, pursuant to that, I'd like to make some suggestions on how to get more funding. The first thing I'd suggest that you never stop trying. Year-to-year funding levels are extremely sticky, so if you can just once get over the hump and get significant money, you should be able to ride that for a while, maybe forever. Request as much money as you think you'll be able to spend in any given year, or as much as you can - the worst they can do is say no.
The second is to ingratiate your team by whatever means possible with any permanent members of whatever boards hand out your money. That has done wonders for us. Believe it or not, my experience is that quizbowl is actually a much more meritorious expenditure of this school's money than most things that request or receive funding. If you can show the right people that this is the case, you will immediately find your fortunes changing. Also remember that control is usually a matter of politics, generally academic and often student, which will inevitably be stupid and petty, so you don't want to focus on any single result, just on the overall funding level.
The third is to aggregate your influence with similar-minded organizations. Contact your uber-funded Model U.N. or your Indian Quizzing Team or whatever else you might have and see if they don't want to throw their lots in with you to go after a bigger piece of the pie. You can maybe make an umbrella organization that represents a lot more people and more diverse interests. Maybe you can even get enough influence to get someone elected to a committee or whatever; you're in funding heaven at that point. I tried this at Illinois with no success - I'm not sure why, but the couple other academic competition organizations I could find (like Model U.N. etc.) had no interest. However, noted best-funded anything anywhere Chicago leverages this strategy and I'm convinced it's the best one.
The last thing I'd like to suggest is that you seek out other sources of funding. Often there are other bodies and offices within a university that have discretionary funds that can be disbursed to organizations. For example, I once attempted to implement a scheme to get money from our very well-funded Engineering Council, which represents the engineering-related organizations and is itself a class-one organization in the above hierarchy, perhaps in exchange for our running a small engineering bowl tournament (unfortunately, this fell though in the end.) For another example, we've been able several times to get additional travel funding for national tournaments from the Office of the Provost. University hotshots like Provosts and Deans eat-up stories like "We're a bunch of smart people wanting to represent your university in a national smart-people contest if we can only find the funding." Also, I assume this point is obvious, but you should run tournaments if you have the facility to do so and make as much of your own money as possible.
Hope that helps,
Centennial High School & Team Illinois 2016-17
Illinois ABT & Fenwick Scholastic Bowl Alumnus