Why can't I send a phone number through AOL Free Mail? Is there a way around this?

I would imagine they block phone numbers to prevent some sort of spam or fraud that was being done using their service.

You could try spelling it out instead or just string the numbers together like this:

(123) 123-4567 is how it would normally be formatted. Instead you could just send this:
1231234567

Or do this:

One, two, three, one, two, three, four, five, six, seven.

One of the formats is bound to work. :)

Use the following table, numbers on the left, and other characters on the right.

1 | l (Lower case letter "L")
2 | 2 (Can Leave as is)
3 | E (Some People Like "E")
4 | 4
5 | S (Capital letter "S")
6 | 6
7 | 7
8 | 8 (Some People like "B")
9 | g (Lower Case letter "G")
0 | O (Letter "O")

That usually gets past most phone number filters

3l5-868-g668

Its not the best looking, but its legible. It is also good for making sure your number doesn't get hit by spam bots when posting publicly.