It's something many stay at home moms fear, that they are hurting their child socially by not putting them in daycare or doing a million activities with them from the time they are born. But it's not something you need to worry too much about, so long as your children do get some social time.
Infants, of course, don't really play with each other, so that early in the game there is little to worry about. But as they grow into toddlers, they definitely need to be around other children, even if they don't interact much until they are 2 years old or so.
One thing to remember is that activities don't have to be formal. You don't have to pay for art classes, gymnastics, etc. Just take the kids to the park. Most times there will be other children there for them to interact with if they like. Or they can just run around and play on the playground equipment.
Especially if there are cousins or neighborhood children of similar ages around, it can be very helpful to get the kids together just casually. Whether you do this by having all the kids play out front, in one family's back yard or have quick get-togethers, this is a great way to ensure that your child does get some social time.
Of course, if your child has spent a lot of time with just you, it can be hard to get them to pay any attention to the other children. You may have to work harder in this case.
One thing just about all children love to do is color. Go to your local home improvement store and buy a piece of whiteboard. A 4-5 foot long piece will probably run $5 or so. This is big enough that several children can color on it at one time. Get the washable crayons rather than markers and let the kids go at it. They may not be playing together as such, but they're certainly aware of one another and will have to take turns using the different colors. There will be arguments over colors and who colors where, but that just encourages interaction and it is up to you and the other parents to keep it friendly.
You should also take a look at what your child enjoys doing with you that could be done with other children. From playing catch to board games, there are many ways to get young children interested in playing with each other. If your child learns to do something with you, they may be more interested in other kids when they realize that they can do that activity with them as well.
Being a stay at home parent does not mean you are depriving your children socially, but it can mean you have to put some thought into the matter. You can't just stay home all day; you need to take your children out to play where they will encounter other children. You don't have to spend a lot of money on this, just some time.