I've been using Affinity Designer and Photo lately, which seems to offer a combination of vector and raster layers using 'personas'. The closest thing to the old Adobe Fireworks which did a similar thing long ago.
I don't do printing though, so it may or may not be suitable for that. You could give the trial a go.
I think you can approach this in both ways, however, I would use a high-res, flattened out (✝) PSD and/or TIFF format, because:
it will cover both print and web usage
it will be less prone to error vs. an AI file with linked or embedded artwork. In my experience AI files are more sensitive to variations in app versions (CC vs CS6 and so on) or in platform (...
You can use either of these two apps. In photoshop you can handle your images quite smoothly and your vactors too, they don't get rasterized unless you convert them.
since it contains images so the end-result for online presentation i.e. on website, mobile display would be an image. It can not be an EPS or Scaleable Vector file.
If the logo contains ...
Definetaly not jpeg's. When I'm applying, I'm attaching my CV and Portfolio seperately in PDF format.
I used to include my CV at the end of my portfolio PDF but that also meant for people to download my whole portfolio and then extract the last page if they wanted to have it printed it out. So it's good for everyone to have your cv separate.
I know it's ...
1- Choose Print (from File)
2- Click on the Rendering Tab: Select (change to) Vector.
4- Click on the General Tab, at the top in "Select Printer": choose the printer to PDF, for example on Windows, I have Foxit PDF Printer (because I have Foxit Reader on my machine).