Can I pay my federal and state income taxes electronically using a Merrill Lynch brokerage without incurring penalty or cost?

If you've set up a cash management account with Merrill and have check writing and/or routing code/account then you could probably do something directly like a bank (not sure what features they offer - that's a question for Merrill).  Or just move the money to a bank account - presumably linked to Merrill - and pay out of there.  Either way you can make online payments using EFTPS for Fed if you have routing/account # and presumably your state has similar electronic payment platform also.  My experience with EFTPS is they will debit your bank very quickly, by next day, state may be a little slower but you should have the money in the account.