I'm not sure what your neighbor fertilizing their lawn has anything to do with your "organic" growing philosophy. Lawn fertilizer gets watered in so will leach down into ground water possibly but it will not flow laterally on the soil surface. There is virtually no chance it will affect your growing operation, even less so since the bags have a bottom. Nothing will leach back up into the bag.

If you still have concerns - which you shouldn't - get some old wood pallets and place the bags on them.

Yes, pallets would work. But if those are too treated for OP, there is untreated wood that would work. Pavers or concrete blocks would work to keep the pots off the lawn too. But anything you put on the lawn will smother or kill the spot under it. Not sure if OP can have dead spots in the lawns or not. If not, the OP is going to need to think about things are are more raised and less of a footprint like metal wire plant racks or stands.

Not sure if lawns means the neighbor is also fertilizing the OP's lawn? If so, the OP might be able to ask that the neighbor no longer fertilize their lawn. If the neighbor isn't doing anything to the OP's lawn, it's fairly unlikely their yard activity will impact the OP's yard. I'm going to work on the assumption that the OP has already asked and confirmed that the neighbor is using synthethic instead of organic lawn chemicals.

You get nitrates from the atmosphere and from rainwater. Just like you get from lawn fertilizer. Oops, there goes my organic garden!

Organic Philosophy? If you're talking about herbicides and pesticides I can completely understand but fertilizers are another story. There is no difference what's so ever in a mineral fertilizer produced by chemicals and a nutrient mineralized from microbes naturally. Any thing you've learned which states one source is better or worse than the other is a fallacy.

Each person has their own views on what they consider organic, and you've seen a couple of those here. I won't get into that discussion, although I do agree with gardengal that lawn fertilizer is not going to leach up into your container plants.

That said, if your plan is to move them around throughout the growing season then your best bet would be to use old wheelbarrows or garden carts, if you have them or if you can buy them cheap. Otherwise I'd be building something similar using pallets (or lumber) and cheap wheels (Harbor Freight has wheels of various kinds/sizes). Bicycle wheels would work too. Grow bags filled with moist potting mix and plants can be heavy and awkward to move around frequently.

Rodney