Do I need a whole new chain for my swingset?

For starters, chain isn't that expensive. If you have any doubts about the strength or condition of the chain, spring for new so you don't have to worry about your kids.

Secondly, adding chain is a snap - or rather, a snap link. Except I wouldn't recommend a snap link in this case though you can find them that would work. What you need is something more like this: http://www.homedepot.com/p/Everbilt-3-16-in-Stainl...

If you decide not to trust the existing chain, here's a good choice to replace it: http://www.homedepot.com/p/Crown-Bolt-2-0-x-30-ft-... It's definitely not the cheapest but certainly not the most expensive either, and is of a sufficient size to be plenty strong to hold up your kids or even you if you so choose, and a nice size for holding on while swinging.

I recommend stainless steel because rust never sleeps. This repair link I linked is plenty strong to hold up your kids so I don't recommend getting any bigger than this; bigger than the one pictured and you might not be able to fit it through the links of the chain you have.

But yes, you can add to the existing chain with no problem.

If you use proper repair links, it will work. Swingset chain is not terribly expensive, but if you really need to save the money, you can add to it.

Adding chain should be OK if you get the same gauge and can completely close the loops. Price the difference, if it's a matter of saving a buck or two, get the new chain, but if more $$, try adding pieces.