I need help recovering vital data from a corrupted sd card formatted as internal memory with the phone off This will be my first post on a stack exchange site, so please excuse any mistakes I make if possible.

[This first paragraph is backstory of sorts; not really important]  

My father's phone's SD card recently became corrupted. Neither him nor I know the reason as to why. I did not know what was on the card, so I made no attempt to immediately rescue any data. A while later, his phone began spontaneously rebooting for no obvious reason after being used for a short amount of time -- again, neither of us know why this has happened. I afterwards found out that this SD card contains vital information relating to the medical treatment of a family member.

Assuming it is possible, how would I go about recovering data from a corrupted SD card which was previously formatted for internal storage without having the phone on? In this case, the phone is able to boot into recovery fine, and is able to actually start up for a short period before it appears to crash.

The phone is a Blu Life One X2 64gb internal storage with a now corrupted 64gb SD card formatted as internal storage.
If the SD card was formatted as Adopted (internal) Storage recovery of this data is only possible if the device was rooted prior the SD card being adopted and you have access to the device in a functional state... If you can't satisfy both requirements then your data is not recoverable, at least not in our lifetime, if fifty supercomputers that could check a billion billion (10↑18) AES keys per second (if such a device could ever be made) would, in theory, require about 3×10↑51 years to exhaust the 256-bit key space to brute-force hack the data. 

If the devices was rooted and functional, it's easy, and that information is available in multiple other questions here. The key component to these solutions is you must access the encryption key from the device, which requires the device be functional and fully rooted, have a Linux computer with a microSD card reader, and a bit of Linux terminal knowledge. 

The rooting must have occurred prior to adopting the SD card as internal storage though, because for most modern devices rooting requires unlocking the bootloader, which also wipes all data in the device including the encryption key.

Without access to the encryption key from the device, the data is not recoverable by any known method. Be aware that even if you happened to get the encryption key somehow and if the data is actually corrupted in the card, recovery of any data will likely be impossible.

Advice for the future: Don't use adopted storage (internal storage) unless it is absolutely required for the device to function as you wish, meaning it would likely have very little internal storage (less than 8GB or possibly 16GB), as adopted storage puts very high read-write usage on the card which can causes it to fail unexpectedly, a situation which is quite common as you have experienced. SD card expansion can still be added and used as Portable Storage (external storage) and can be used to store pictures, documents, music, and other files and data, this usage is not encrypted and can be read in any compatible device.