Is there an in universe explanation as to why Walter
recorded his plan
onto video tapes
before ambering it,
rather than using DVD or video file.
It seems to be very problematic to retrieve them and they seem to be very prone to degradation when removing them.
There isn't a technical explanation. The use of 1970s technology is just a quirk of Walter's character. The 1970s were Walter's hayday, and he's somewhat stuck there. That's when he was young and made the most of his scientific breakthroughs. That's why much of his equipment dates from that period.
Other than the explanation that Walter seems to be attached to older technology, as evidenced by his rather large vinyl collection (seen in a few episodes, eg: 4x08), there could be another reason.
From a purely technical standpoint, tapes like Betamax (Walter explains it's a Betamax, not VHS in 5x02) could've been a better medium to store this than a DVD or a video file. Tape degradation happens mainly because of humidity, which I presume is not a factor in amber.
But magnetic tapes degrade organically, so even if it does degrade, you can still make out some of the footage (this was portrayed somewhat accurately in the show). But with digital media, it's (literally) binary: if you've lost data, it's all gone.
If you have used VHS tapes you'll remember how the damaged parts were still somewhat watchable, but in a DVD even minor scratches can cause it to skip parts of the video completely. It would have been a problem if important parts of the plan were completely lost.
I think that Walter meant for the tapes to be somewhat of a challenge for whomever found them. He could not have predicted when they d be found, it could be decades. So, say in 2070, it would be very difficult for the finder to watch the tapes.
