sendmail is used for sending mail, so not relevant here. Fetchmail can do this for you and can be told to download mail from specific accounts for individual users, so it does exactly what you want. However, it is not the most user-friendly of programs to set up. Getmail may be a better alternative, it comes with better organised and more readable documentation, as well as example configs. Each user's .getmail directory contains a file for each account they want to download from and their crontab contains one of more lines to run these configs as often as you want.
"Insanity: doing the same thing over and over again and expecting different results." (Albert Einstein)