"I see that the yield is 4.63%"

Maybe, maybe not. The dividend is never the same two months in a row:
http://www.etfconnect.com/select/fundpag...

"but shouldn't the yield change based on how AGG trades throughout the day?"

That's a bit more information than almost anybody would need, let alone use.

"How would you go about calculating the current yield?"

You take the annual dividend and divide it by the current price. In this case you'd have to assume an annual dividend, since it's different every month. Using this month's dividend, .3270, the annual rate would be 3.924. 3.924 divided by the current price, 101.03, results in a yield of 3.88%.

Pick a different month's dividend, multiply it by twelve and calculate for yield and you'll get a different number.