What is a budget friendly way to fix my old flooring in my kitchen?

Vinyl flooring can be inexpensive, depending on what you choose. Your photo appears to show structural damage to whatever is under the floor. You could squeeze silicone caulk into the cracks and wipe off the excess, but addressing the problems and replacing the floor is the only long-term fix.

I've used nail polish to repair really small chips in my composite tile floor by mixing colors to get a match. I used nail polish remover to clean up the excess and level it. If you try this method, try the remover in an inconspicuous spot to ensure you don't do further damage.

Also, dheck YouTube to see if you can find a solution for this particular issue.

https://www.youtube.com/results?search_query=repairing+chipped+vinyl+flooring

Go to your nearest Habitat ReStore and look around. You can get discount home repair and interior decorating materials, from flooring to chandeliers to furniture and electronics. All proceeds go to Habitat for Humanity, and that is 100%.

During the depression of the 30's women painted their linoleum a solid color and then after it dried they used a big natural sponge and dipped it lightly into a contrasting color and made a print . I guess you could get artistic and make the sponge marks into a central pattern - etc.,

I agree with Ken, looks like there is some structural damage. If you are just looking to cover it cosmetically temporarily, peel and stick vinyl tiles are great. They're cheap, go down on top of existing vinyl (as long as it's been cleaned really well) and they even have some that look like wood

There are tubes of laminate repair in various colors that dry hard. I would fill in the holes with the repair. Let dry and see if you need to reapply. You can build up various coats. This should fill in and keep stuff from falling through the cracks. Its kind of like colored custom caulk. It's not costly. Nail polish also, but I do not think it would be a good filler for the crack.

another person above has pointed out that lino is a very cheap way to lay new flooring in your kitchen (and if required service rooms e.g. laundry, bathroom etc) and you can get some now that look like all the fancy floor boards. If however laying new floors is not in your budget. Have you considered just buying a suitable glue (ask at your hardware store what is best). Sticking down the crack and other damaged areas. Then covering the entire thing with a big mat suitable for the room. Remember people don't judge you on your home if they are in your home they are there to see you, not your floor.

Just make sure there is no asbestos in or underneath it, if you choose to remove it. Asbestos was used in or under vinyl flooring until close to the mid 80's. I unfortunately have some, so I will just be covering it up.