Resolution:

when HTTP_REQUEST {
    if { [string tolower [HTTP::uri]] starts_with "/v2/" } {
        # Set Artifactory server information
        set ARTI_PORT "443"
        set ARTI_PROTO "https"
        set TIMEOUT 900
        # Parse original host request
        scan [HTTP::host] %\[^.\].%\[^.\].%s REPO DOMAIN TLD
        # Rewrite the uri
        HTTP::uri /artifactory/api/docker/$REPO[HTTP::uri]
        # Set request headers
        HTTP::header insert X-Artifactory-Override-Base-Url "$ARTI_PROTO://[HTTP::host]/artifactory"
        HTTP::header insert X-Forwarded-Port "$ARTI_PORT"
        HTTP::header insert X-Forwarded-Proto "$ARTI_PROTO"
        HTTP::header insert Host "[HTTP::host]"
        # Update the idle timeout
        IP::idle_timeout $TIMEOUT
    }
}

Basically, take a docker command such as docker pull docker.artifactory.test/busybox, where docker is the repo, busybox is the docker image, and artifactory.test is the server name.

You want it to rewrite the uri to /artifactory/api/docker/<repo>/<image>, or /artifactory/api/docker/docker/busybox

Take a look at the following for more info:
https://devcentral.f5.com/articles/irules-101-16-parsing-strings-with-the-tcl-scan-command