Unfortunately, it looks like ZipZap, the billing company for BitInstant that shows up on your payment slip, only accepts payments at drop-off locations and does not accept online payments.
If they did accept online bill payment, there would be another radial option for online payments there.
Are you doing something other than selecting the 'Pay Bills' option on their site? The Moneygram Express seems to be the wrong option, since the forms are invalid (it looks like they are only for use with packets that are purchased in brick and mortar stores), and the other options are for sending money to be physically picked up in another location.
I did successfully use btcquick.com this morning. I had to register and upload a photo id, then wait for them to validate me.. but that wasn't too difficult. They used Google Checkout and charged a credit card. The fee was 7.5%, which is a little higher than the ZipZap to Bitcoin address' 3.99% fee. However, ZipZap charges a $3.95 USD fee for processing the cash transaction, so depending on quantity this may be cheaper. It was easier than digging my vehicle out of the snow.
One thing I couldn't find on either BitInstant or btcquick is how the BTC price is determined. I'm not sure if it's a weighted average, lowest ask, or highest bid based on values at purchase time on some specific exchange, or if they have some predefined value.. I should have kept an eye on that and checked around a bit once I received the btc so I could get an idea... but the price seemed fair. I'm going to look around both sites to see if I can find this information.
