Yes you can put raw chicken into a slow cooker and the slow cooker will cook the raw chicken fully and slowly.
The slow cooker is designed to cook everything that you place into the slow cooker including the raw chicken and as long as you allow enough time for the slow cooker to slowly cook the chicken it will fully cook and be ready and safe to eat.
For example if you put some raw chicken into your slow cooker and go to work for 8 or 9 hours then the chicken should be fully cooked after you return home as long as the slow cooker was on the highest setting.
The same goes for if you put raw chicken into the slow cooker and then go to sleep which is also safe to do and I do it all the time.
By the time you wake up the raw chicken that you put into the slow cooker should be fully cooked but you can also just turn the slow cooker down to the keep warm setting and then go to work or wait until dinner or lunch to eat the slow cooked chicken.
You can brown the chicken first if you want too and it will slow cook much faster but putting the raw chicken into the slow cooker without browning it or cooking beforehand is okay.