Please, please, please, be very careful what you do with your violin. If it's over 50 years old, then there is a very real possibility that it could actually be much older than you know, and you could ruin a priceless treasure if you aren't careful. The only way to know for sure is to look at the label inside.
Now, regarding the peg, there are a number of options available, including buying new pegs for the instrument. However, there is also ways to repair the peg holes in the violin by gluing shavings of wood in the peg holes to make them smaller. Very thin shavings of wood will suffice. If this is a choice, then use only " Animal Hide Glue" to carry out this repair. Animal Hide Glue is natural, and is what is used on all older violins, and most new violins even today. It's strong and durable, yet 100% reversible and will not damage the instrument further.
Another important consideration is that the owner bought the violin used. That is a tell-tail sign that the instrument is much older than its suggested 50 years. This is especially true since the woman never used, nor played it at all. That is a clear indicator that you MUST be careful what you do. Take it in, have it assessed by a professional Violin Shop. Let them examine it and give you an accurate assessment for what to do with the violin. People have found and bought old violins in antique stores and in thrift shops and have discovered their old beat up violin was a priceless antique like a "Stratavaria", many reported to be worth 10's of thousands and even millions of dollars.
The fact that the pegs were loose indicates great wear and possibly very old age. It also suggests that the original owner probably and likely used the instrument a lot, until the pegs got worn. The rest was left to time, from the wood drying and shrinking. If it's in very good condition, then it needs to be assessed and examined by a professional.
For someone to just say "Get some Pegs, Dope" is not only crude and very rude, but shows a significant amount of ignorance and insensitivity to something like this. Ignore them.
Regardless of its true age, this is something you purchased because YOU wanted it. Don't let someone bully you into doing something that you may end up regretting. If it is older than suspected, then it would be worth the investment to have it professionally cleaned, restored and repaired.
I am new to violin repair, but learned a lot over the years. I've seen very old 300 year violins that seemed impossible to repair, completely restored and made playable again. Loose pegs are a minor problem, but I am innately suspicious about the true age of your instrument. Please, let a professional look at it, ask them to tell you what you have. you will know immediately if it's a rare artifact when they look at it. Violin shop keepers have a keen eye for something rare and will start drooling, shaking and their eyes will light up when they see a collector's pieces. They will know instantly Yours, could be one of them. Good luck fixing the pegs, treat your violin as a treasure, because it already is one. It's yours, and it's already over half a century old that you know of. That in itself makes it a treasure. Thanks for sharing.