Yes, there are already people running Monero full nodes with Android using GNURoot (which doesn't require a rooted phone, despite its name) as suggested by villabacho.
The proof of concept can be tested easily according to the steps listed by binaryFate:
- Install GNUroot Debian
- Launch it, and type the following into the terminal:
apt-get update && apt-get install wget bzip2
wget -O monero.tar.bzip2 https://downloads.getmonero.org/arm
bzip2 -d monero.tar.bzip2
tar -xf monero.tar
monero/monero-wallet-cli (use --daemon-host for a remote node)
If you have enough storage and want to have a better privacy, download and store the blockchain on a SD card. Otherwise it may be more feasible to use a remote node, which is faster and easier.