Is there anything in the rules that say you can have your army use vehicles from another faction?
I use Harlequins and I always thought about trying something new and using some Eldar vehicles like a War Walker or a Wraithknight. I figured it would work since Harlequins and Eldars are the same race. Or with human races like a Militarum Tempestus army using a Space Marine Land Raider or two.
Is this possible?
Yes and no
In the current edition, all armies (except unbound armies) are made up of formations. For example, you might take the Combined Arms Detachment from the basic rules, or any other formation from your codex. Which army you are using is basically determined by which faction your warlord is from. You can take formations from other armies and books, however, as long as you follow the rules for allies, located in the rule book. (I don't have my book with me to get a page reference).
So, you could take a Harlequin Combined Arms Detachment and also take an Eldar Warhost, which would give you the ability to take a lot of Eldar stuff, as long as you fulfill the requirements of the formation. I'm sure there are other Eldar formations that would have less stringent requirements, as well.
Additionally, you could take an unbound army, which has no force org requirements, but also grants none of the special rules associated with formations (no objective secured or warlord trait reroll, for example). In that case, you can take exactly whatever models you want. You would still have to follow the rules for allies, though, which can make movement or deployment difficult if you pick armies that do not like each other.
Yes, this is entirely possible under certain circumstances
You see, the core rules have a section where you can make your war-band. One of the options that you have is taking multiple detachments. Since there are no real restrictions on transports on what they move, unless they are "Allies of Convenience" or worse, so long as they are paid for, you can have them.
However in order to make this work RAW, you need to build in a specific way. The only ways that make sense is to make the transporter and the riders their own Combined Arms Detachment, or downgrade so that the riders are an Allied Detatchment (see the following image).
In either case, this means that you need at least an HQ and 2 troop units (the troops can take dedicated transports) in order to use that factions' transports (between 3-6 depending if the factions' Heavy Support options have transports). The only other stipulation is that since the units are split between 2 formations, the troops of the rider faction cannot start the game inside of a transport faction vehicle (however they can embark inside of the transport when you start your turn)
Ynnari allows you to put any infantry that falls under the same detachment as a Ynnari leader into any transport in that same detachment. Herald of Ynnaed - check it out.
If the armies are Battle Brothers you can share transports, for example have Howling Banshees in a Dark Eldar Raider. Or put Harlequins in a Wave Serpent.
If you want Formation benefits (like Objective Secured) you'll need to field 0-1 CADS, plus any number of Allied Detachments or Formations. Many tournaments will require you to field at least one CAD. If it's a friendly game you can field whatever you want, though of course you won't get any benefits, and only Battle Brothers can share transports. If you go that route, do check the allies rules, as you may need to setup different faction units (and will probably want to keep) a certain distance away from each other.
