No, if you can't install LR on your computer, then you also can't share edits between a tablet/phone and that computer.
Why can't you upgrade the Mac?
The MAC is almost 9 years old. I think it has something to do with the fact that the OS is unable to upgraded past Lion. I know I need to upgrade to a newer one but the cost is a little much. I have Lightroom 5.7 on the MAC and Lightroom mobile on the ipad pro. Just have not figured out the sharing or if it is even possible.
If your Lightroom 5.7 is not the Creative Cloud version, then sharing is not possible.
You may be able to subscribe to the Creative Cloud, it would install LR 5.7 Creative Cloud version on your computer (because your operating system won't allow Lightroom CC 2015 to be installed), and then you could share between computer and tablet. I don't know if this will work because I haven't tried it. You might want to try donwloading and installing the 7-day free trial of Lightroom CC and see what happens.
Thanks DJ. I will try that and see what happens. :-) I appreciate the help.