Q: Do photos in my library update even if the program is not running?

Yes in some cases - why do you ask? And for very technical questions like this you need to tell us what software you have as the answer may change with different versions

for help see Writing an effective Apple Support Communities question

LN

Yes if you are using iCloud Photo Library - BTW uploading to ICPL is very slow and does not continue if the computer goes to sleep so set the computer not to sleep and let it run night and day - it took me about ten days to upload 7000 items

LN