We just purchased a 55 yr old home that had vinyl replacement windows installed about 8 years ago.

The the seam between the two panes of glass on the largest window in the living has come apart a bit and all these little beads, which I think are there for insullation purposes, keep falling out.

I don't know if this will be an issue or not. Has anyone seen this happen before? Can I repair it myself or do I need to send it out?

 

Unrepairable.
Time for new windows.

 

is the glass sweating in-between the panes of glass? Any water stains? Fog?

Yes I have seen it happen even in new windows, as long as the seal is good and not doing any of the above, your fine.
But it will more than likely start doing the above, if it hasn't yet.
Start saving a little extra cash, time to buy a new unit soon.

 

The little beads are "desiccant" which are in the spacer between the 2 panes of glass. The desicant absorbs any moisture that might be in between the panes. This is not normal for the beads to fall out of the spacer. If the seam in the spacer is opening and allowing the beads to come out, it was either defective when the insulating glass unit was made or has opened because of stress caused by the glass not being properly supported in the frame.

As glasman2 has said, leave it alone until you see moisture between the panes then it'll be time to replace the insulating glass unit.