Is it possible to care for a puppy while working full time?

It definitely can be done. That’s why it's important to have a safe and secure place where you can kennel your puppy throughout the day.
If you have the ability to go home and check on them over lunch, even better. That's better for them, but that's basically what it takes. Give them the chance to use the bathroom just before you leave, put them in their proper bedding with things they can't chew up, destroy, choke on, etc. Try to check on them once if not twice during the day to let them out, clean up, and just to make sure they're not getting themselves into anything that they shouldn't

It is. People do it every day, although it’s a big commitment. You certainly want to make sure that you can provide that puppy with the appropriate time once you are home from work because they will need training. They are going to need to be able to be let out during the day sometimes.

So, If you have a tiny puppy, you have to keep in mind that you may need to be going home during your lunch to let them out just because they won't be able to hold it as far as needing to urinate or have a bowel movement. So, that may be something to consider as far as getting an older puppy or an adult dog instead.