No, this is not possible. A partition mounted read-only is read-only, meaning you can't write your tar.gz file to it.
That said, do you have access to the network on this device? I think you should be able to rsync them to a separate computer and generate the tar there.
If you really need to generate the tar archive on the embedded computer and it is connected to the network, then you might try using netcat: https://nakkaya.com/2009/04/15/using-netcat-for-file-transfers/
In tar, you would add the "O" or "--to-stdout" flag to the command. Pipe the output of that to netcat.
On receiving end, run:
nc -l -p 1234 > out.file
On embedded end, run:
tar cvO /partition | nc -w 3 [receiving IP] 1234
Note that netcat isn't encrypted, so this would only be advisable for non-sensitive information or on your local network.
Another possibility is to mount a remote drive using, for example, NFS or sshfs.
On the embedded computer:
sshfs email@example.com:/remote/computer/directory /mnt/remote
tar cvf /mnt/remote/files.tar /folder/to/tar
If you're trying to tar up the root partition, make sure the remote folder is empty so it doesn't get a bunch of cruft in the archive.