There are alot of ways to achieve this, what fits best will depend on your requirements. The following is very simple and just executes shell commands on different jenkins-agents.

If those vms are connected to your jenkins master as agents, you are free to checkout your Dockerfile and build on one agent (or master), then switch agents and launch your image there.

Along the lines of the following snipped

def image = 'image123:latest'

node('build-vm') {
  stage('Checkout') {
    checkout scm
  }

  stage('Build image') {    
    sh "docker build -t ${name} ."
    [...] 
    sh 'docker push ${name}'
  }
}

node('vm1') {
  stage('Run image on vm1') {
    sh 'docker run -d ${name}'   
  }
}

node('vm2') {
  stage('Run image on vm2') {
    sh 'docker run -d ${name}'   
  }
}

node('vm3') {
  stage('Run image on vm3') {
    sh 'docker run -d ${name}'   
  }
}

Ideally, you would use the Jenkins Kubernetes plugin in order to execute jobs on any remote VM Jenkins agents, where kubectl has been installed.

That allows a Docker in Docker build: from "Building Docker Images inside Kubernetes" written by Vadym Martsynovskyy.

You also have other options, detailed in "Pipelines With Docker Alternatives" from "Joost van der Griendt's CI/CD Knowledge Docs!", using either:

• externals nodes (with the JENKINS Amazon EC2 Plugin): not your case
• Maven JIB, with JIB allowing you to build optimized Docker and OCI images for your Java applications without a Docker daemon
• Kaniko, which is a tool to build container images from a Dockerfile, inside a container or Kubernetes cluster.
• or Img, which is a standalone, daemon-less, unprivileged Dockerfile and OCI compatible container image builder

The main point is: build docker containers with Jenkins is often done in a Kubernetes setup context.