Can I recover deleted images from Capture One's proxy files?

If we are talking about a catalog, the answer is: Yes and no.

Full disclosure: I have no idea if there is a chance when using sessions (never used one) or if there are some highly-specialized programs that might do the trick - at least my Google-Fu did not reveal any ground-breaking program for the job. Also, consider trying some recovery tools - avoid writing to the disk where your RAWs were stored before to increase your chances!

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I tried it with all file-types inside <YOUR_CATALOG>Cache. IrfanView, which I consider to be a very open-minded program, can open the *.cot - files and claims that they are JPEGs with wrong endings - however, they appear to be thumbnail-sized only (around 300px on the long edge).

I then tried to open all other file-types (*.cof, *.bc, *.cop) with 7-Zip - also to no avail.

XnViewMP failed with all those formats. FFmpeg, on the other hand, was able to read the *.cot and opens up *.cof as well - though it only shows what appears to be random noise.

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However:

One solution is to make a screenshot of the preview in Capture One - depending on your settings, this should far supersede the 300px thumbnails IrfanView can open, but it is a very time-consuming process.

Unfortunately, Capture One cannot export offline media only with their thumbnail available, as this Q&A suggests:

While offline, you can still make adjustments and organize images as Capture One does not require the raw file for these actions and will use the cached preview file. However, it is not possible to zoom in past the preview size, process/export or create an LCC from an offline file.

If you're working in a session- No. The preview proxies (.cop files) are in fact rendered image files (since they are the previews you see in the Viewer panes. Without source files, the session can't reference anything to process.

If you're working in a catalog- Somewhat. Since the source files are gone, the images will be marked Offline. You can, however, leverage those preview files (which are within the catalog package) and process to the QuickProof JPG format. This format essentially takes the preview files and uses them as the source data to generate JPG files. This data is limited to the size you have set in Preferences > Image > Cache Preview, but generally it's big enough for web.