Went to Diamond City to buy rad away from this vendor and cannot find it on my pip boy.

Should be listed under the Aid tab od your inventory section. There's also a quick option on the stat screen as well (The one that displays Vault Boy)

Make sure you completed the transaction. Even though you went through the vendor inventory and selected all the items you want to buy, you need to confirm the sale at the end to make the full exchange. Check your cap count: Did it change? If not, you didn't complete the transaction.