Visit your local Home Center like Home Depot or Lowe's, or visit a flooring Retailer like Floors USA, or something. Explain to them what's going on. Essentially, you'll need to 'clean' the area under the vinyl that's coming up so when you re-apply some adhesive, it will take hold. You DO need to most definitely CLEAN the 'edges/seams' of the vinyl (ie., the 'edges' of the vinyl that, essentially, 'touch' each other.) THen, you'll need the APPROPRIATE Seam Sealer.
If you can't get the necessary material from the aforementioned locations, contact Armstrong directly. We have exactly what you have for flooring. We have 3 'sheets' in our kitchen but our seams are where the 'edges' of the cabinets are; NOT down the middle of the kitchen! They should be located (in the future) where there's the least amount of traffic if possible. Obviously, w/o seeing your kitchen and all, it's hard to tell if the job was done as best as possible. In any event, the next floor you get, be sure to talk to the installer about THIS current install (and its problem(s).)
My best to ya and hope this helps.
Jay J -Moderator
PS: Our floor is almost 5 years old. Our seams are still good, thank God. I just dry-mopped it this AM. DO keep it clean because cleaning it on a regular basis obviously keeps the seams clean. It isn't 'abnormal' for seams to come loose. It's just their nature. Given you've had yours for 10 years, it sounds (on the surface) that the install was a good one.