Similar answer as the other, but perhaps with a slightly different direction. You didn't provide much information about your community, so this will be a fairly general answer. Your fundraising should be related to the purpose of your community. If you are able to provide high value experience related to the purpose of your community, it should be possible to find funding sources from a variety of means. None of these are exclusive. You can mix and match.
In addition to creating and focusing on a purpose, try to find a few other dedicated people who help you get started. These people will be the beginning of your elite core of members.
1. Ask for donations
Similar to another answer, but how and who to ask is part of the key. There are a few sources of money you can ask that might be relevant:
- Grants - if your purpose has a societal good impact, you might qualify for grants from government or philanthropic organizations
- Crowdfunding - Kickstarter or other crowd funding sites are a possibility. You might not get operating costs, but you could likely get startup funding for design and acquisition of software and services. This is also highly useful because it literally allows you to start building up your community as part of the crowd funding campaign. A key point here is to have a clear deliverable. The quote here from Kickstarter explains better:
Kickstarter can be used to create all sorts of things: art and
gadgets, events and spaces, ideas and experiences. But every project
needs a plan for creating something and sharing it with the world. At
some point, the creator should be able to say: “It’s finished. Here’s
what we created. Enjoy!”
- Ask your users - even Wikipedia asks for donations. They put out their costs and run a fund raising drive every year or so. A key thing to understand here is that your best and most elite users are the most likely to give you the biggest donations. So this is why you should have a plan for deeply engaging a few people rather than expecting a mass audience from giving you a little help from everyone.
2. Create a subscription service
Here you need to figure out a tier of service or capabilities or features related to your community's purpose that your users might be willing to pay for. This likely will require significant build-out and investment before its good enough to be a premium experience. It will also take a while to have enough critical mass of subscribers to be a good revenue source. The thing to be careful of here is not to restrict access to features that actually help build your community. For example, not letting people post to forums or engage in online chat would keep them from meeting other people in your community, and thus fail to help you find more potential subscribers. Again, here, look to your elite users. Is there something they want or need that perhaps they'll be willing to pay for? They're your most likely subscribers first.
3. Affiliates, referrals, sponsorships
Similar to the affiliates suggestion in another answer. Are there products or companies that would clearly be of interest to members of your community. Ideally it should be related to its purpose. Perhaps you are a community of Poodle lovers. Then providing information for dog grooming supplies might be a value added affiliation page that companies could pay for. You could also ask for purchase referral fees for every successful purchase thats made. Such programs exist for books, and for apps as well. Sponsorships are similar, but a bit more wide open. If there's a company that members of your community will appreciate, try getting them to sponsor your community in exchange for branding or banner ads. Except here you control what's displayed, not a 3rd party ad service, and you get a higher share of the sponsorship. Obviously this gets better as you get more members.
4. Extend off a paid service
You didn't talk about this in your description, but its worth mentioning anyways. Many communities are collaborative experiences related to a paid for product or service. This could be commercial, or non-commercial. Some examples include support forums for products, social media for sports leagues, fan clubs, etc. If your community helps drive a physical experience or sales of a related product, the business should provide at least some funds for startup.