Wherever I search, I can't find any resources on how to run Docker on an iOS device.
Is iOS a valid host OS for Docker?
Docker is an automation tool, which facilitates deployment of applications using Linux feature called Linux Containers. Docker provides neither virtualization nor emulation environments. As of today, running Docker containers requires using Linux (support for other operating systems, like Oracle Solaris and Windows Server is in the development phase).
Docker for Mac and Docker for Windows are separate products that run a Linux virtual machine inside the host OS and provision containers inside that machine.
There is no way of running a Linux virtual machine in iOS, hence iOS cannot be a "host for Docker".
Besides operating system, even if you ran Linux on iOS, you would need to also consider the CPU (that is also true for Raspberry Pi recommended in the comments under the other answer). Most available Docker images use binary packages for Intel/AMD architecture, not ARM limiting the use cases compared to running it on MacBook.
iOS would be an extremely poor choice for docker use. Setting aside that code needs to be signed to run on iOS, these general design considerations raise problems to implement docker on iOS:
The iOS design makes running persistent server tasks difficult and in the end, wouldn't it make sense to run docker on a swarm of networked raspberry pi and just orchestrate the setup and configuration from iOS?
For that I run prompt and ssh into Linux or Mac hosts to set up docker instances and control them.