The intention is that I would extract the visuals from the file and hand them off to someone to re-insert wherever they need them
That is what CC Libraries are for:
What are Creative Cloud Libraries? | Adobe Creative Cloud tutorials
I'd ideally like a vector format that is easiest for others to work into/edit.
If they are already vector and you drag them into a CC library, they will be stored as vectors. If you have bitmaps, you will need to covert those to vector, if that's the format you want to use. You can do that in Illustrator.
should I copy them into Adobe Illustrator and save them as EPS format
Save them as a native .ai file. EPS files are outdated.
EPS – the Zombie among file formats
Save as PDF (export or print). If your elements are vector objects they will be vector in PDF (except object that can be flatten, check this by flatten preview before export). PDF can be opened later in illustrator, coreldraw or some another "vector" program.
I would recommend, either a CC Library or select them, copy them paste them into a new INDD file (nothing else), export (not print) to PDF/X-4. This can be placed in any INDD file as linked file.
NEVER use EPS in 2016. EPS is an old outdated file type. Avoid also any flattened PDF as it is PDF/X-1a, PDF/X-3 or any PDF which is created via print.