Ooof. Technically, yes absolutely a router can do that.
However you'll need more than just a plunge router to accomplish this without a genuinely ass clenching experience fraught with peril* . You'll need a two sided router edge guide aka mortising jig, and you'll also want a 7/16" router bit. Those won't be cheap.
Your best bet is to use a cordless (or corded) drill with a 7/16 drill bit and shop made jig that's basically just a 1-3" block (depending on depth required and drill bit length) that you drilled a nice straight hole into. Then you can screw a top board onto this block so that the hole lines up with your intended destination when you place it on the workpiece.
Clamp the jig onto the workpiece securely, and let it guide your drill bit.
*your router will jump on you if it's not secured and it's hogging out a decent amount of material - even if you go slow, once you waver a tiny bit in the hole it'll jump and the best case scenario is that you ruin your workpiece. Also, your RPMs are very important. If you've got the speed set too low, or engage the router bit before it gets up to full speed, or stop the motor before you've disengaged the bit, it'll jump on you.