Hey
From experience - InDesign export to JPEG is not great and the general better way is::
Export to PDF and then convert in Photoshop
Converting the InDesign to PDF then converting it to a JPEG in Photoshop can sometimes be more manageable and less resource-intensive.
You can set up a photoshop action to suit your needs.
If you want to keep working with InDesign that's no issue.
First thing I'd check is InDesign's Temporary Files
InDesign I think might be creating temporary files while exporting.
These can build up and take up space, particularly large files/high-res artwork.
You could try checking the Temp Files as you are working on the file, usually, these temp files should automatically removed when you close InDesign, but maybe this not happening for you.
To check this you could try close and reopen InDesign which may free up disk space as you have noticed I believe.
It’s also a good habit to manually check and clear temp files if they linger after a session:
On macOS, temp files are usually stored in /private/var/folders/.
On Windows, look in the %temp% folder.
(these might be hidden folders)
For a test you could Lower the DPI or Quality - and test your files
150dpi or 200dpi then if you're happy with them you could then export the final files at high quality once they’ve confirmed everything is as expected.
Optimise Linked Files
Since the original artwork is high-resolution, consider whether downsizing the resolution of the linked files, for the purpose of working and exporting within InDesign, might reduce the strain on the hard drive. If they are working with unnecessarily large files, that could potentially take up additional space.
This would involved making a duplicate copy of your images and resampling them before putting them in InDesign - which is going to take up more disk space.
Check for Available Disk Space
It sounds like the drive is running low on space during export. They can check how much free space is available on their hard drive and try freeing up additional space. A minimum of 50-100GB is recommended for large projects.