Is there any way to get into the warbird community without being rich or otherwise connected?
Of course the answer is "yes" here, but there is obviously nuance.
I'm not being snarky when I say you're not going to find anyone who is going to pluck you out of the crowd and lead you by the hand to a cockpit of a waiting aircraft. Getting involved as a pilot is going to be transactional, and you're going to have to bring something to the table. You can bring experience, money, or volunteerism to the transaction, and generally it is going to require some level of all three in combination.
Option one is, you can buy your own entry-level aircraft, a'la Stearman, PT-19, PT-22, etc., and not rely on joining up to an organiztion. This is doable on an airline pilot budget, and aside from initial checkout, you can play to your heart and wallet are content without anyone else. There are good numbers of other folks who own and fly 'em, too, so there's a community to fly/socialize with if you so desire, and ready-made mentors/help to learn new skills, maintain, etc. Plus, this experience is valuable if you later want to move up to more complicated warbirds that are beyond your personal budget.
Option two is the one with the nuanced and complicated answer. I look at it a lot like the hunt for a career flying job we've all gone through, where there are skills you're going to have to spend your own time and money to develop (e.g. the basic tailwheel flying experience), then you're going to have to network and go check out the people and organizations that are keeping warbirds flying (e.g. museums, CAF, etc) and join. Those orgs are generally looking for people to join and help, but you're going to have to make the first move with them. After that, it depends on what they need and what you're bringing to the table. The personality of each org is different, and what they're looking for is different. What you bring to the table for each one may differ, too. Again, the combination of experience, money, and volunteerism is going to be different for each one.
The money part of the combination can be very low, but it is probably going to require you to do a lot of volunteer work even if you are bringing experience to the mix.
It probably won't be a one-shot deal either way. I've been associated with a dozen or more private owners, private museums, or nonprofit orgs that own, restore, and fly warbirds over the years. Each of them had their own objectives, and sometimes my objectives and what I had to offer them merged up. Sometimes they differed, and I moved on to something different.
I'm currently primarily involved with the Commemorative Air Force (in two different units in two different locations), but that's also after I personally spent a decade or more as a "never CAF'er" after the negative interactions I had with it in the 80s and early 90s. Of course, I'm also bringing an entirely different palate of experience to the mix today than I did 30 years ago, just as the CAF has a different organizational goal and mindset today from then.
Orgs change, people change, circumstances change.
Hope that's not too obtuse. I can quote some numbers if you're interested in knowing an example of the costs, and I'm sure you're aware that there are people right here on JC who are initial networking contact points associated with some different points-of-entry into the warbird world who are willing to help.