You can scale all your windows, using the window size in the Options menu.
Or, if it is just the chat window, you can use the command /windowscale chat X, where X can go from 0.65 to 1.99.
i am honestly not sure if that is possible ive never tried it but best bet is to go to menu and options and double check under graphics section maybe there is something there.
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Also, before scaling your UI windows you may want to try resetting the resolution for the game to match your computer monitor. It sounds like you've chosen the wrong resolution. Otherwise the Chat UI should start out pretty big and easy to read by default.
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Otherwise the Chat UI should start out pretty big and easy to read by default.
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"Easy to read" is pretty subjective, actually. My monitor is set for 1280 x 1024 and I played in Windowed mode. I think the text default for that is 12, maybe 11. I always bump mine up to 13. Can I read it if it's smaller? Yes. But I don't consider it easy. 
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My monitor is 1680X1050 and that's the resolution I set the game for. I was able to size the windows from the Options menu. Thanx.
I can read the chat text okay but the chat window is small in my resolution and it's really hard to see the little letters on the top edge which indicate which chat channel. Any way to fix this?