How can I make home-made veggie burgers stay together so I can grill them?
I second the recommendation for Lukas Volger's Veggie Burgers Every Which Way. I've been working my way through it and have made several of the recipes so far, all of which have been good. However, I would say that not all of the recipes in the book work well for the grill. 

I've been doing everything vegan from the Volger book with mixed results. Many of the recipes use eggs, but you can do them with just breadcrumbs and water as the binder. They come out pretty delicate that way and are better in a skillet and then baked, as Volger recommends. His tofu and chard burgers use some tofu that has been browned then pureed as binder. This works ok, but the burgers came out quite delicate (but still delicious).

Silken tofu pureed in the food processor gets a thick, creamy texture that could also be used as a binder.

Eggs are by far the most common binder as they may be the best edible binder there is, since they are basically liquid protein, but obviously not vegan. 

For delicate burgers, I use this cooking grid on the grill, which works great. You can find something similar at most kitchen stores.

Another binder option is steamed rice, prepared like you would for sushi. It gets sticky and makes a great binder, though they'll still be a bit delicate.  I've been making the beet, brown rice and black bean burger recipe from Volger's book. I add bbq sauce and a paste of pureed rehydrated prunes, similar to the veggie burger at Hillstone/Houston's restaurant and they come out really great. They work fine right on the grill, though refrigerating them overnight makes them a little sturdier.

Another option is adding vital wheat gluten and water, aka raw seitan dough. This is what is done in the Veganomicon black bean burger recipe. I tried these straight on the grill also and they stayed together perfectly, as well as any commercial veggie burger. However, they were a little bland and could use some more seasoning.

I haven't tried any of these in burgers, but some common binders in vegan baking are peanut butter, apple sauce and Ener-g egg replacer. I bet peanut butter would work well in a burger with flavors that work with peanut, like Thai flavors or chipotle.

Falafel-style burgers also don't need any binders, they stick together pretty well and can be grilled, though you may want to use a grill grid or something similar.

Lastly, you can put a cast iron skillet or pizza stone on the grill - it's common now for grilled pizza and will work for delicate burgers.