Are we still Researching the Foundations of Math?
Of course we are doing research in the foundations of mathematics.
We want to understand the necessity of ZFC, and in particular the role of the Axiom of Choice, in our mathematics. This means that we need to understand how mathematics in ZF could look like.
We want to understand the influence of set theoretic axioms on the mathematical universe, and vice versa. For example, if we wanted to add the assumption "Every set of reals of size $<2^{\aleph_0}$ is a null set" to our axiomatic system. Is this going to add a contradiction? Can we maybe prove this? What are the consequences of such assumption, or its failure?
For this we need to develop tools for studying the set theoretic universe, for studying independence in set theory, and then we need to study the "actual mathematics" one can do when assuming this statement or its negation.
For the independence results, we have forcing and inner models in the study of set theory, and we study forcing axioms and cardinal characteristics of the continuum, and other combinatorial axioms with these tools, as well as their influence on the rest of mathematics.
We may want to study other foundations. Perhaps you want to formalise your proof into a computer program, and perhaps your approach makes ZFC an unsuitable foundation. Maybe you simply don't want to accept the Axioms of Power set and Choice. Maybe you reject the law of excluded middle.
I don't know you. The point is, some people do these things. Some people want to understand how mathematics looks like in other foundations, as well as the interactions between these foundations and ZFC. Examples would be Homotopy Type Theory, intuitionistic logic, Quine's New Foundations, category theory, and more.
All these things, of course, make the study of foundations of mathematics vibrant, deep, and interesting. And we have yet to begun discussing topics like philosophy of mathematics.