How to estimate the prevalence of a trait when those with the trait are more likely to respond? In a certain country, at most 1 % to 4 % of the population has a specific hobby (as per an independent annual survey with results in the same order of magnitude for several years); this is pretty much the only reliable knowledge available. My interest is in finding out connected the hobbyists are to the wider hobby scene. This can be quantified in any number of ways: whether they follow hobby media or not, how many people they talk about the hobby with, do they discuss the hobby online, are they a member of a club, etc. The particular way is not relevant to this question, I believe.
The only feasible ways of reaching the hobbyists is more likely to reach the more connected hobbyists. It would be possible to use online surveys, clubs, or maybe entry surveys to a couple of events (after the current corona situation eases up), for example, or maybe even word-of-mouth starting from a largish seed group. Online surveys spread through relevant communication channels are the most likely mean, maybe augmented by the others. But all of these would predominantly sample the active population.
Does there exist an established methodology for counteracting this strong bias?
My naive approach would be to assume some distribution with a few parameters (generalized power law, maybe) for how connected people are, and then assume that their probability of answering is a function of how connected they are (with some parameters), and then do parameter estimation. But this would involve lots of strong assumptions; they would be educated guesses at best, and hopefully better than nothing.
Of especial interest here is the number of hobbyists who are poorly connected, and thus also unlikely to answer. High amounts of uncertainty are unavoidable, but any means of mitigating it are welcome.
(My own background is in mathemtics, not statistics and especially not statistics in English. Please excuse and fix any misuse of statistical terminology, as well as edit to add relevant tags.)
Further background on the specific problem (in Finnish): https://ropeblogi.wordpress.com/2020/02/28/roolipelaajien-maara-ja-verkostoituneisuus/
 A: One approach would be to create an a priori stratified sampling scheme or template for all hobbyists based on a set of preselected criteria. These criteria should be structural (e.g., demographics or geography which are not easily manipulated) and include the factors discussed in your first paragraph from published sources regarding 'how connected the hobbyists are to the wider hobby scene', assuming such information is available to you. 
It's likely that you will have only summary information (marginals) from these published sources, for example, the percent of a published sample who are members of a club (making some numbers up, 60% yes and 40% no). 
These marginals can be worked with but it's a little tricky, statistically speaking. 
Iterative proportional fitting (IPF) also known as raking is a technique that's been around for decades. It's been used for adjusting the results from a survey to correct for the kinds of bias you've identified. It will enable you to create a weighting scheme to apply to the survey that you will eventually field and adjust for the bias you've identified based on a priori expectations. 
Cross-validated has 46 questions devoted to raking and 22 questions about IPF. 
A: I would suggest a variation of the non-response sampling methodology.
In the case of random selected samples that refuse to respond to, say a questionnaire, there is a follow-up smaller random sampling design,  referred to a doubling sampling, involving perhaps, direct calling or other incentives to respond, to gather information. 
In the current context, similarly a double sampling of those with the hobby. A follow-up interview with the rare few that have the hobby could suggest important details as to where, when and how they participate in their hobby, for example. This may suggest an improved path to contacting the group and in estimating its percentage in the parent population.
A: A really good question, and I like user332577's answer, but I think you can find some middle ground between making strong off-the-cuff assumptions and accepting a biased estimate without adjustment. For one, it may be logical that limited income increases barriers for hobbyists to be "connected," so it might make sense to apply the hobbyist percent from higher income classes (if you have the demographic crosscut) to lower income classes. Moreover, if there were similar "propensity to connect" estimates for some other hobby, it might be reasonable to apply those to the hobby in question here.
