How do you build a team that is already present? The least you want is that the members understand each other. That is an essential first step in team building.
This can be done by an external assessment. Companies are not equipped to do this by themselves, nor should they focus on these kind of activities, which occur sporadically. And there is another benefit by using a third party.
But first search for a simple online organizational assessment. In such an assessment you will have your team respond to a series of "difficult" questions about their organization (either focussing on the current situation or a new one). Take for example, this question and think how you might respond:
There is a new solution available on the market. Your organization is already involved in a project with a fixed deadline. However, the new solution will give you a lot of extra features and will bring more innovation. What is your response?
Different team members will respond differently to such a question. Reckon what will be the outcome if you have fourthy of them.
Normally the result of such assessment will be measured in various dimensions.
Than you should present the results to the individual members. And additionally you could set up a workshop to discuss the outcome of the group. Not to learn the individual answers (that will reduce the incentive to participate), but focusing on where the team is aligned and on what issues the team in out of balance.
The discussion in such a workshop will serve as a teambuilding event. You could do this with or without a predefined benchmark. A benchmark will help if there is a strategy set, otherwise you can leave this open if you want input for new possible directions.
Internet is a powerful means of providing such alignment tools. And they are there in different options. The benefit of using a third party is that they can be independent.