This shouldn't be a coaching question. The real focus should be on how to effectively collaborate with your vendor as a business entity and with their people. It's likely that some critical step was missed when chartering this project, and those initial assumptions should be carefully reviewed to ensure that the project itself is operating as intended.
How to Proceed
How does one circumvent the fact that the teams he opted to coach consist mostly of external consultants?
This is likely to be a problem with the way your company is composing or managing "teams." Teams aren't just gaggles of people thrown together; a team must be composed of people with common goals and a compatible working style. Just because you have both internal and external resources allocated towards the same objective doesn't make them a team; on the other hand, using outside resources doesn't inherently prevent teaming, either!
The underlying X/Y problem isn't about agility, or the lack of it. The primary questions should be:
- Who is managing the outside consultants or the engagement within your organization?
- Who is responsible for meeting the project's key objectives?
- Are the goals of the consultancy and your company's management team contractually or pragmatically aligned?
Embedding your internal resources onto a vendor's team is fine, but doesn't give your organization any control over day-to-day processes. Likewise, bringing in a consultancy to deliver a service doesn't obligate them to follow your internal processes unless it's specified in the contract.
Your company isn't powerless. The golden rule is "He who has the gold makes the rules." If you want a different process or a different organizational or contractual structure with your vendors, you can do that. That's not a question of training or not training them, though; it's a much more complex issue of how you engage with a vendor, and what controls you put in place to manage the processes and outcomes for a particular project.
Before you spin your collective wheels on the training issue, step back and ask why you want these teams to be agile. If they're supposed to be internally-managed teams supported by embedded resources, and if how the project integrates within the overall organization intrinsically matters, then by all means inspect-and-adapt your team management and process framework. Otherwise, you're probably focusing too much on process and not enough on goals or outcomes.