No, you cannot play blackjack with a no deposit bonus. Most online casinos will only permit you to play video slots and scratch cards with a no deposit bonus. There aren't any reputable online casinos that offer a no deposit bonus that can be used to play blackjack. Very few online casinos even offer deposit bonuses to play blackjack online. To offer a no deposit bonus for blackjack would be a risky move for the casinos.
If you love playing blackjack, then you'll need to play the game in guest mode, or register for a real money account and make a deposit to play the game within an online casino. There are some RTG online casinos that offer special blackjack bonuses to new players, but these bonuses come with hefty wagering requirements. Casino bonuses aren't typically offered for blackjack because it puts the online casino at a bigger risk.