No, you can't pick up the daily rewards automatically. The only ways to get the reward access is to buy a reward shrine for your house and use it there, use the reward shrine in the adventurer's guild, most temples and depots, or purchase the reward access from the store which will instantly pick up your daily reward from the location your at. Bottom line, you have to be logged in and not logged off to take advantage of the daily rewards. If you miss one day, you can contact customer support if you were having issues but not guarantee they can help.
Also here's something copied from the FAQ on Tibia.com if your having problems with the daily reward for future reference:
You picked up the daily reward on 2 following days but the reward streak was reset? The daily reward must be picked up between 2 server saves to raise the reward streak.
Examples:
- character picks up daily reward on Monday 1 hour before Monday's server save and then 1 hour after Monday's server save = reward streak rises
- character picks up daily reward on Monday 1 hour before Monday's server save and then on Tuesday 1 hour after Tuesday's server save = reward streak reset
- character picks up daily reward on Monday 1 hour after Monday's server save and then 1 hour before Wednesday's server save = reward streak rises
- character picks up daily reward on Monday 1 hour after Monday's server save and then 1 hour after Wednesday's server save = reward streak reset
The shown waiting time to pick up your next daily reward is longer than 24 hours!
The time of your computer is not set correctly. Please correct your computer's time.
Source: https://www.tibia.com/support/?subtopic=gethelp&entryid=158