Your system requires an Internet connection so that it can communicate with the EA servers; all of the game's save data is stored there and EA seems to be a firm believer in the "Always Online" method of anti-piracy.
Without an XBox Gold or a Playstation Plus subscription you'll be denied access to any feature which requires interaction with other players online. This means that online multiplayer battles and online Co-Op in the form of Garden and Graveyard Ops won't be available, but both of these features have reasonable approximations using AI teammates and opponents. The Solo play options won't be as challenging since the AI doesn't come near the skill level of actual human players but you still get to experience the main bulk of the game and you can get a level of customization in both your own AI team and your AI opponents.
The part where you'll probably feel the absence the most is going to be with the Trials of Gnomus. The Trials involve acquiring special keys for a special area where you undergo some fairly rigorous tests of your skills as a player and succeeding at all four of the Trials rewards you with the ability to unlock two new Character classes which are not available by default.
This part of the game relies heavily on earning Rainbow Stars which can only come from playing events three times a month in the Mystery Portal which sits inside Town Hall. The problem you'll experience with "offline" play is that the Mystery Portal is exclusively online multiplayer and online Co-Op and doesn't grant you any solo option. While they do allow you to exchange normal Stars which you can earn from Quests for the special Rainbow Stars the exchange rate is stiff at 7:1 which means that without access to the Mystery Portal it will take months for an offline player to earn the same number of Rainbow Stars that an online player can earn in a weekend and quite frankly you may lose interest in the game before ever unlocking the two hidden character classes.
Now none of that matters if your son just wants to goof around with a silly game about Plants and Zombies at war with one another with all sorts of oddball variations of each character class but if you think he might be bothered by knowing that he's missing out on a part of the game then it might be worth looking into a subscription so that he can join in.