I am a part of a group which all have the same boards. However, there is not a "button" or "link" on my board that allows me to add a card to it. What could be wrong with my account? Our admin on the account wants everyone to be able to add a card to their boards so that everyone can see each other's boards and cards. Please explain what might be wrong.
You may be a member of the 'Organisation' but not explicitly a member of the 'Board'. You can probably browse to the Board, but not make any changes, add cards, etc.
Once you are a 'member' or 'admin' of the Board, you'll be able to do all these things.
In summary, you need to be added to the individual boards as well as the organisation.
With the team board open, click on the team name next to the board title. Click on "View Team Page" and make sure the team member is listed under the members tab. When verified go back to your team board. Next, on the top right of your team board just below the profile picture select the "Show Menu" link. At the top of the Menu, it will show members for that board. The Admin must add everyone from the team that will need to add or edit cards. Select "Add Members" and you should get a drop down list of team members. Once added the "Add" option will appear at the bottom of their card.