How can I make Zap synchronize with the Bitcoin network?
To run the wallets you mentioned, you will need to also run a bitcoin node (bitcoind or btcd, most likely), and an lndnode as well. This information is listed in the README files on the pages you linked to. 

I believe there are lightning network wallets that work without a local full node, but I don’t have experience with one so you’ll have to dig a bit to find one that works for you. NOTE: without running a node on your machine, you will be trusting someone else’s node in order to use the lightning network.