Yes! (But not at the high-quality one gets from a proper sound booth.)
Unless you have an appropriate audio recorder in your place of residence and subjects to record are currently co-isolating with you or have their own research-quality microphones, lab-quality acoustic recordings will not be possible.
Instead, it may be possible to:
Use online recording tools, such as Zencastr
Adapt your research question to be about perception
Explore existing corpora of acoustic recordings
OSCAAR hosts a number of high-quality corpora of many languages
Focus your research on robust acoustic landmarks (e.g., vowel formants) and use moderate-quality recordings
While it may not be possible to perfectly calibrate headphones remotely, there are ways to conduct 'headphone tests' (e.g. instructions tailored for the Gorilla platform) to ensure your participants headphones are adequate for the task. Contact the Lab Manager to discuss options.
However, there are a few online and freely available corpora you can request access to.
Please discuss alternatives with the Lab Manager and your supervisor or colleagues to brainstorm ways to adapt your research question to the available technology.
For visual world eye-tracking, mouse-tracking is one possibility, although there might be ways to implement coarse-grained eye-tracking through webcams. The platform LabVanced offers a state-of-the-art webcam eye-tracking algorithm that will be able to produce relatively high-quality gaze data, remotely.