I go through so many of these annoying plastic bags. Unfortunately, not many places recycle them. Are there any ways they can be re-sealed?
Use a manual impulse sealer. It consists of a linear heating element specifically for sealing bags in the fashion LEGO parts bags are sealed by the factory. Narrow sealers can be purchased new online for as little as $12-15. It's helpful that LEGO parts bags are relatively small.
I use these devices to reuse bags for various non-LEGO parts storage. There is a dial on the front to adjust the heating cutoff. Depending on the material and thickness a different heating time is needed to ensure a good seal without damaging the plastic.
Here's a video demonstrating its use: https://www.youtube.com/watch?v=iH-SbjYNzQc
One way of resealing the bags is to use a laminator. Most laminators aren't designed to seal anything other than lamination pouches so you'd want to be particularly careful using them to re-seal LEGO bags.
No matter how careful you are when you open them, most polybags tear slightly and leave rough edges that cannot be fed easily into a laminator.
I suggest cutting a straight line with scissors or a ruler/box cutter, just below the original seam.
Start up the laminator to get it warmed up and fill your polybag, I don't recommend you fill it up more than 1/2 or 2/3 of the bag as it can be difficult to feed the laminator.
Don't let it run too long as it will be quite difficult to remove (as shown in my example, I left it about a second too late, the second time I fed it into the laminator), it doesn't take a long time for it too seal completely and it takes a few tries to get it work well.
After the bag expands, it should be airtight and you can turn of the laminator, and remove it when it is safe to do so.