Of course, an external party will not come and do this exercise for free, there is a cost associated with this, but the main advantage is that you know upfront what the cost will be, what the timing is, what the deliverables are and what you can do with this output by when.
When you do this internally, there is also a cost, although in most cases, this will be hidden and, true, easier to put into the budget than an external invoice, but still the cost is very much there.
But if you have someone dedicated working on this, this person is fully accountable to this project and if you have someone doing it “whenever there is time” then the cost is also there, but it remains hidden.
The most important downside of this last option is that the cost may be very high, since it will probably take very long for this person to come up with a clear deliverable, which leads to suboptimal customer experience for all this time, you won’t be able to improve your customer interactions or your customer support as long as this exercise is not done
This may indeed be much more expensive than having an external party come in, make a clear, end to end customer journey map, complete and not hindered by any internal politics, filters or sensitivity, giving you very fast the so needed input to make your customer experience good and personal, making sure you as a company, needing the love of your customers so badly, can attract, nurture and grow your customer relationships.
And yes, as an external party it may seem I’m not completely objective on this subject, but having worked for a long time as an internal employee at several companies, I have tried to map the customer journeys making use of internal resources and I know first-hand where it all went wrong and why it never was finished. So if I were to do it all again as an internal stakeholder, I would definitely hire an external party to do this for me and I know the customer experience of my customers would have been a lot better a lot sooner.
So if I may give you this piece of advice: either put someone senior, that can move cross departments and cross silo and that can work on this in a really dedicated and filter-free manner, or hire someone external.
And please don’t think that it is enough to map the journey for your department only, because, unless you are a very small company, your customer is most likely to interact with lots of different departments in your company and they all need to be mapped into one big picture. Only if you do that, you can make your customer journey work. Approaching this from one silo will not work and may even make it worse.