My two cents
In short
A good customer journey map, or service blueprint, is often one built with many people.
Here a few tools that you can use to let others fill a customer journey map:
Use specialized tools like Smaply, Custellence, UXPressia, or TheyDo .
Use a spreadsheet app like Excel or Google Sheets
Use good old sticky notes
Start with a template
Many of these tools have templates (created by the companies themselves or users) that can help you to get started:
Smaply and alternatives
Smaply is a tool that is specialized in the creation of customer journey maps.
So you can create a new customer journey map and add a few collaborators in it. Then a few different people can work on the same customer journey map together.
There are a few other tools that are specialized on the creation of customer journey maps, like Custellence, UXpressia or TheyDo.
Miro and alternatives
Here, there is a bit of work to do on your side to create a templates that people can just fill.
But the good thing is that today many people have already used a digital whiteboard and therefore it might be pretty easy for them to create a customer journey map in such tools.
Most of these whiteboard tools also provide templates that you can use to get started in a quicker way.
Excel
If people aren't used so much to digital whiteboards or these kinds of new collaboration tools. You still can go back to a good old Excel file.
The advantage of a good Excel file is that people don't have to use another tool. Or have a new license.
And even if the IT firewall is crazy in their company, they still usually have access to Excel.
Here again, you will have to do a bit of work to create or use a template that people can fill very quickly.
The added bonus is that people can work offline on their Excel document and then send it to you via email.
Sticky notes
If the people you're working with are definitely not digital natives and hate the computer, good old sticky notes work perfectly.
In such a case I would recommend that you take one of the big walls in your company building where a lot of people go through.
And then ask people to add their thoughts about the customer journey map with a new sticky note.
You'll need a bit of instructions on how people should use that wall and the sticky notes that are provided, and add a few examples of what you expect so that people already see what they should do.
Obviously, don't forget to put a few pens with it so that people can write.