What research methods do you use when the budget is very small? Do you rely solely on online surveys due to the budget constraints or is there a better way?
One of the best solutions for this situation is benchmarking. Maybe you can't pay for user research but, you can use others' researches. Find similar products and start to study them. sometimes they publish their researches on their blog or some related blogs. don't push yourself to solve problems just like them but in this way, you can be aware of problems that you need to solve. Just try it. If You're working on a new product or can't find a similar one I suggest you find some persons similar to your persona and start to contact them and talk with them about the product. you can also talk with stakeholders and ask them about their experiences and users' thoughts.
The most costly part of user research is actually most likely to be the time and effort required to analyse the data and synthesize the insights, assuming that the research method chosen is appropriate and the right tools and resources are in place for the collection, management and presentation of the data. Otherwise it will be a costly exercise regardless of the budget (research debt is a REAL thing).
As much as you may want to save money on tools and resources, generally the free versions of tools have other limitations, and reducing the amount of users for research and testing also can result in having to do more redesign later on if you don't collect enough data.
Hence, I normally think about designing the same methodology but scale it to suit the budget, rather than picking the method based on budget. I think that's the best way to go about it.