Our group is too small to train visitors or (unenrolled) students in experimental methods, and there is no infrastructure here in the psychology department for visitors in theoretical linguistics. On rare, but rewarding occasions, we have welcomed visitors who actively contribute to experimental or modeling work on language for shorter or longer visits. If you are interested, please see the Princeton website for information about formal and financial requirements for visiting graduate students or visiting postdocs. We cannot accept visiting undergraduates except in very unusual situations.
For German postdocs or researchers no more than 12 years since PhD, it is possible to apply for funding from the AvH Feodor Lynen Fellow from the Humboldt Foundation, Germany; link.
(If you are instead interested in visiting the linguistics program at Princeton, contact firstname.lastname@example.org for information.)
I apologize that there are such limited avenues for extended stays here in our lab.