Can a pulsar have an axial tilt close enough to $90^\circ$ to hit us with both beams for two pulses per rotation?
Yes. One of the features that helps this scenario is the gravitational bending of light close to the neutron star, which allows for a larger fraction of the surface to be visible at a given time.
Many pulsars have double-peaked light curves, for example here's the light curve of the Crab Pulsar. In the pencil beam model these peaks correspond to the beams from the two poles.
According to Annala & Poutanen (2010) "Constraining compactness and magnetic field geometry of X-ray pulsars from the statistics of their pulse profiles", the ratio of single-peaked to double-peaked pulsar light curves puts a most likely value for the maximum inclination of the magnetic axis relative to the rotational axis of 40°±4°.