Why not? Is there some guideline from the AWS or similar organization on this?
I'd always heard this, stories that MIG welded chromoly would stress fracture at the weld quite easily, and I used to take it as gospel. So a few years ago I decided to do my own non-scientific experiment. I MIG welded some scrap chromoly tube (made a 90° mitered corner), without pre-heating or normalizing, just let it air cool, put it in the vise and beat it with a sledge for a good 10+ minutes, then I put a 3-4' cheater bar on it and wailed on it in both directions for several more minutes, to no effect. Having believed what more experienced folks had told me I was honestly surprised that I could not break it. So then I gave up breaking it and cut it in half... and everything looked fine, then I was skeptical.
Having read and learned a bit more since then, I now believe that MIG welding 4130 chromoly is perfectly fine and safe, as long as you let it air cool at a moderate temp, and don't quench it in water or oil.
Also, as better evidence I know that both Cub Crafters and Maule (possible others too) MIG weld all their fuselages with 4130 chromoly steel. Granted they have better control and facilities then some guy in his garage, but it must be possible and perfectly safe or the FAA would not have certified those airframes. There are hundreds of Maules and Carbon Cubs that get beat around bush flying daily, and I've never heard of any stress fractures at the welds on them. In fact Cub Crafters claims that their fuselages are stronger then the original Piper ones.
What's really neat is some of the new technology like "friction stir" welding, no gas, no electric arc, just friction from a tip spinning very fast under pressure. Makes a beautiful flat weld with no filler.
Well, when I finish building my Bearhawk, I would like to show in my builders log that all welds were crafted properly so my DAR gives me an airworthiness certificate. That coupled with the coin that I shelled out for my EAA class on welding leads me to believe that the pros know what they are doing.
It is physically possible to perform MIG welds on 4130, but if the joint is cooled too quickly, it will be a brittle weld. There is no other way about it. It is the nature of the steel because of the carbon content. The "41" denotes a low alloy steel containing nominally 1 percent chromium and 0.2 percent molybdenum (hence the nickname "chromoly"). The "30" denotes a carbon content of 0.30 percent. If you heat 4130 to the point where the carbon comes to the surface and then cooled too quickly (even a drafty garage is too quickly), it develops Martensetic qualities (a crystalline structure typical of steel containing carbon that has been heat treated) the weld will become brittle. If this happens, it has to be tempered just like tool steel or it will develop stress fractures and possibly break. An O/A flame weld has less tendency to do this
When welding an airframe, especially the clusters where there are lots of tube joints, there is a tendency for that area to become hot enough that the joint requires normalization. A MIG or SMAW welded airframe, if performed by a homebuilder, will not pass the DAR's inspection. This much I know.
Having a TIG welder at home would be nice, especially considering all of the welding that I do on my Wrangler and the parts that I fab for it, but I cannot justify the cost, especially since I dumped money into a lathe and mill. In fact, TIG welding is becoming the method of choice for homebuilt aircraft enthusiasts and when I am ready to start assembling my frame, I may consider buying one. For now, all I that I have been welding are the steel parts for the wing assembly (control surface/flap linkage/stick, etc) which are easily done with my O/A setup. Besides, torch welding has an almost zen-like quality about it

Some of the older builders swear by it. I have seen some older welders repair construction equipment in the field with nothing more than a portable O/A torch and a coat hanger and still make the weld look like a stack of dimes

As for the friction welders? I could only wish
