Scientific realism is a thesis that is concerned with the relation between scientific representation and the external world. Linguistic idealism is concerned with the relation between linguistic representation and inner experience.
Therefore the two thesis are not directly related. It's common nowadays to think that scientific representation is not linguistic (and also that scientific experience is not simply founded on inner experience). But even assuming that scientific representation is linguistic, the idea that language "shapes" experience says nothing, a priori, about its relation to the external world.
Having said that, there is an epistemological link between the two ideas. The idea that theoretical language shapes scientific experience, put forth by Kuhn or Feyerabend, brings suspicion on the idea that experimental results tells us anything about the external world. Experimental results would be some kind of theoretical artefacts, at least in part (note that this is very far from consensual that this is the case). This entails that the realist idea, that our theories correspond to the world, cannot be justified by their empirical success. Therefore anti-realism. Note that this kind of view assumes semantic internalism: that meaning is wholly "in the head".
Back to linguistic idealism: it has indeed affinities with scientific anti-realism, but a few things are needed to bridge the gap:
- Scientific representation wholly derives from linguistic representation
- Scientific experience wholly derives from subjective experience
- Meaning is "in the head" (perhaps implied by linguistic idealism)
To my knowledge, phenomenologists and existentialists do not take scientific knowledge very seriously (Merleau-Ponty would be an exception, as he takes input from experimental psychology). I would say that generally, they are anti-realist by default, but that the question of scientific realism is not their main preoccupation.