What can be used instead of cheese to help a grilled sandwich stay together? Grilled sandwiches like paninis almost always have cheese in them, which is important both for taste and structure. The cheese works like glue to hold the bread together and keep the ingredients in place. You can make one without cheese, but it's logistically difficult (the insides fall out while cooking) and I find it a little empty without the texture filling all the gaps.

Are there some substitutes that will provide the same material benefit without being cheese? I've used roasted sweet potato to great success in quesadillas, but it's a very strong flavor that won't play well with a lot of sandwich types.
If you find roasted sweet potato to be a strong flavour, that makes things harder. Mixed roast vegetables, including some roast sweet potato or squash can work well, and the flavour of one vegetable doesn't dominate. 

A rather thick hummus is also good (it softens a little on warming). In fact, roast veg and hummus is probably the vegan panini I see the most. Peanut butter should work, but I find that when heated it can become unpleasantly sticky (depending on brand; other nut butters that start stiffer may be better).

All these options have their own flavour, but so does cheese, even if very mild. The trick is to work with it, not against it.