You have two "sub-questions" here.
Can I use a cord end replacement kit for a cord in use in my home?
Yes, absolutely. In many cases the repaired end may be of better quality and safer then a original end. Not so say that the original end is unsafe, but, for example you can get a "fatter" plug that covers the socket better and helps keeps your toddlers fingers out of the socket.
There is no governing body (IANAL, I am in the US, may vary based on location, other legal disclaimers as needed) that can tell you not to use a repair kit. There is a "governing body" that certifies that the kits are safe to use.
From Lowes is a good example.
It's fatter, which helps keeps toddles from touching the prongs, and "Meets UL FedSpec WC596" which means it has been deemed safe to use.
Are there any legal issues when using a replacement/repair kit on an extension cord?
In your home no. In a work environment, maybe, but generally no.
So lets focus on a work environment. Lets say you have a coffee pot, plugged in via extension cord. Now Bill, the receptionist comes along and cuts the extension cord. Bill can not just stick the repair kit on the end. He is not qualified. But he can call Bob, the electrician, and Bob can come out and stick the end on. That said, Bob charges $190 an hour, and he already has to come out (Bill can't replace the extension cord either, cause it's on a coffee pot). So do you pay Bob the electrician $190 or just ask if he has a spare cord laying around he can let you have. Bob will probably just give you an extension cord, and plug it in for you, because your a long time customer and it only takes a second, and if you can wait till Thursday he will be in the office anyway. But he will charge you to install the plug end.
Which is now better. Paying Bob $190 to install a plug end, or just getting a new extension cord.
There are some situations (usually high voltage, or high amperage applications) that a plug end is not allowed. But even these are few and far between.
When using a cord repair/end replacement kit properly, it is perfectly safe and legal. In the "business world" it gets more complicated, just like many other things, but when done properly is still generally legal.