No, there isn't really a "standard" to do this.
As far as bills go, each company has their own billpay system. Small businesses pretty much all take credit or debit cards now. If they don't you pay by check. I pay my property taxes with a check because if you do an electronic payment with their system they want you to pay like a $3 fee. Buying a house requires you to get a cashier's check.
You can do "billpay" online with your bank, but if their systems aren't "hooked up" to the other party's systems the bank will mail a physical check to the other party. None of it is instant or even fast.
Last time I changed banks (about 6 years ago) I decided to experiment and not order any checks. Every person I've encountered who needs a check will wait for them, so I just add them to bill pay and have the bank deal with printing and mailing them a check. So far I haven't run into a situation where I was cursing myself for not having actual physical checks.