Former (long ago) blanket maker and repair-er here.
There are a lot of older, simpler, cheap machines that can do the work, but you kind of have to buy one to try it! Actually sometimes you can take stuff you’re trying to sew (say, several layers of woven nylon along with some polypropylene webbing) to a commercial sewing machine place, and ask them what they have that will sew it. You can find these places in the ugly downtowns of big cities, IE, LA, New York, Chicago, etc. Don’t judge a machine by its looks, let it show you what it can do.
If they’ll let you test-drive an old machine you may be able to find something that’ll do the trick for not much money. Look for upholstery-type machines if you can, they have higher clearance for materials under the foot usually. And the simpler ones are usually better- you don’t need to be able to embroider a hemline of baby ducks, you just need it to sew forwards, and a reverse lever is nice too. Make sure it can handle heavy thread too, and one that takes pre-wound bobbins is super-handy. If you can find one with a walking foot is great, that means that both the bottom teeth (called the feed dog, in the base plate of the machine that the needle goes through to catch the bobbin thread) and the top foot (that fingers-looking thing that sits on top of where the needle enters the fabric) move with gears and tiny teeth on the parts that help pull top and bottom layers of heavy fabric through the needle area at the same time. Generally… that’ll be a machine designed for the heavy layers in upholstery, and sometimes for denim work.
Biggest issue with a lot of machines, and any repairs really, is the filth from blankets. You really, really want to wash them before you let them near any machine that has oil in it! Horse hair and dander + oil makes for a horrible mess and gummed-up machine. And you want to oil your machine frequently; old commercial machines sometimes even have an oil bath down by the bobbin that continually lubricates the moving parts of the needle area and adjacent.
Yes, needles are important- ask the sewing machine place, and buy a bunch of them. Be careful sewing through really heavy stuff, too- if the needle breaks off, and you have your li’ll face close to the needle, trying to see what you’re doing with a size 84 blanket crammed through the throat of the sewing machine… shards from a shattered needle can really, really hurt you… especially your eyes. Beware!
Another thing to think about is WHY the blanket failed. A lot of repairs are because blankets are often made from stupid designs, with, for example, front straps that are not well-reinforced on the inside of the blanket. Buying better blankets (and a lot of blankets are way better than they used to be) is often a good value move. And adding inner reinforcements, in the form of something strong that hopefully doesn’t fray like lightweight leather (or burning the edges of nylon) to make patches can help keep the same failures from recurring.
Nylon is lots stronger than polyester, and both are stronger than cotton of the same size diameter. Denier, by the way, is a measurement of the fabric’s thread thickness, not anything else. One denier, legend has it, is the size of a single strand of silk (notoriously strong for its size, but not seen much in horse blankets!) You also need to know what the FIBER is, not just the denier. 600 denier Nylon is a lot stronger than 600 denier polyester. Can you tell my Mom taught textiles?
Anyways, try to mend clean blankets, be creative about improving the design or strength underneath straps that seem to fail repeatedly, and don’t overlook ancient homely sewing machines that might just love a job sewing through layers and layers of horse fabric! My friend who is a legendary chap maker has been making them for 50 years on an ancient straight-stitch Singer that was already 50 years old when she bought it!
