There are two meanings of "foundational research".
If you just mean mathematical logic (containing computability, set theory, model theory, and proof theory), there is a lot of ongoing research in those fields. Of course the cutting-edge results are usually technical, but the same can be said for every other well-developed area of mathematics. Nobody would read a paper by Galois and think that it is reflective of cutting edge work in algebra, or read work by Cauchy and think that is it reflective of current research in analysis. Similarly, it's a mistake to read papers in mathematical logic from the first half of the 20th century and think that they are reflective of current research in the field. If you want to see current work, you could look at the Journal of Symbolic Logic or the Journal of Mathematical Logic, both of which are well-regarded research journals in the field.
Sometimes "foundational research" is used in a different sense, to mean work that is supposed to provide some sort of philosophical foundation for mathematics. For better or worse this is not the direct aim of most researchers in mathematical logic, although they are happy if their work does help provide insight into foundational issues. The idea that there is some "universal foundation" on which all of mathematics is built is much more difficult to defend in light of what we currently know, compared to what people knew in 1900 or 1930.
One recent example of the interplay between technical research and foundational insight is in algorithmic randomness. This field was initiated in the 1960s, but in the 2000s there was an explosion of new work, much of which is documented in the recent 855-page book Algorithmic Randomness and Complexity by Downey and Hirschfeldt. While many of the results appear technical to outsiders, they do provide a much clearer foundational picture of randomness than anyone had in 1995. They do this in the modern style, by deeply exploring and comparing the mathematics of multiple notions of effective randomness.