Great question! Many people think this is just a personality thing that can't be coached, but the truth is that compete level and hunger can be coached.
The key would be engage them in drills that promote aggressive, initiating play, and positively reinforcing the behaviour when it happens. Hopefully when it happens the player gets a good result, which will reinforce the behaviour. The player will probably like the result, which will further motivate them to play aggressively in order to seek out that result. At the same time, doing this in practice will build the player's confidence in that style of play, making it more likely to show up games.
Confidence issues and being scared to make mistakes could certainly be a possibility here, as some players are very good at hiding these things. It never hurts to ask your players about what they feel their style of play is, and starting a harmless, curious discussion about what that player wants to be like and tweaks to their game they could make in order to get there.
Hope this helped!