Is it possible/advisable to use a VCR/VHS to record to get an analog sound on a budget?

You can try.  I haven't kept up w/ VHS technology, but used to the analog stereo tracks were recorded linearly while the video was stripped horizontally.  The sound quality on good decks was pretty good, but I doubt if you would get great analog sound like a tape deck.  Smaller track width and more consumer electronics in the front end.  Still, you could try.  They were higher quality than cassette - which isn't saying much.

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I used to try it on VHS HiFi (is that what the super duper version was called? and while freq and noise spec wise it was good I could never get the deck to handle transient well. (god awful 'spitting breakup..) Had it cleaned, shop aligned' etc. Keeping the levels down helped some, but bottom line it was like walking on eggs.
IIRC and understand it it was their 'stiching slices of audio back together that had it's own bag of downsides, not like 'tape at all perhaps.

A "hi-fi" VCR uses frequency modulation to convert the audio spectrum to a much higher-frequency spectrum and it's then recorded along with the video signal using a rotating head.  Without going into detail, the rotating head approach has the same effect as if you moved the tape across a fixed head extremely fast, and it's a clever way to get the frequency response needed to record a video signal.  The downside is that because the analog signal undergoes frequency modulation, all desirable analog tape effects such as saturation, distortion, wow & flutter, etc. no longer apply.

VCR's also have a fixed head that's used to record a linear audio track.  Because the linear speed of the tape is so slow, this track has high distortion, bad frequency response, poor S/N ratio and a handful of other problems.  Analog circiutry in consumer grade VCR's is quite poor and recording to this track in an effort to sweeten up your guitar sounds will likely result in disappointment. (unless edgy and brittle is what you're looking for)

My suggestion is to forget about the VCR idea.  When people talk about the desirable effects of tape and analog electronics, they are usually describing sounds that are achieved using professional grade equipment.  If you told me you had a 2" 16-track Ampex multitrack laying around and you wanted to use it to sweeten up your guitars, I'd probably tell you to use it on everything!

You can probably find a few different free plug-ins that will do a surprisingly good job of giving your electric guitars an analog sound.  Take a look at the Bootsy plug-ins.  I've played around with them a little bit and they seem to work pretty well for adding some analog character.