Can I use the MAX3420E in a self-powered peripheral?

Absolutely. In fact, the MAX3420E has features specifically intended to support self-powered applications. For example, in a self-powered application the peripheral needs to know when the device is plugged into a powered USB port. The MAX3420E's VBCOMP (VBUScomparator) pin is connected to VBUS, and routed to an internal comparator that provides an interrupt request at plug-in (VBUSIRQ) and another interrupt request at disconnect (NOVBUSIRQ). As another example for the MAX3420E, a bit called VBGATE (VBUS gate) can be set to automatically disconnect the D+ pullup resistor whenever VBUS is detected to be off. This is a required USB specification.