Strictly speaking, your team is singular, indeed. I understand that you do not want the reader to give the impression that the whole team acts as one entity, though.
I can see two obvious ways to keep yourself happy and stay out of the hands of the grammar police. One is to make use of English' habit of reusing verb forms and make your sentence a bit longer:
Our team of economists who are at your service and available by phone or email will be happy to respond to your questions.
This way, who are...email is a subordinate clause referring only to economists, so the grammar police will allow a plural, and strictly speaking, will be happy is singular, and referring to your team. It is exactly the same as the plural form, though, so maybe you fooled the grammar police.
Another option is to keep your sentence as it is and tell the grammar police that you have used the singular they in your second sentence. They can be used in a sentence to replace the sometimes awkward he or she (or the even more awkward he/she). That you would usually refer to the team as it and not he or she, you can answer with "our team is human", or "if they is gender neutral, it should include the neutral gender".
I'm pretty sure that if the grammar police insists on arresting you, you still stand a good chance in grammar court.