HI MD, welcome to the forum. Doesn't look like anyone here has answered you unless they have PM'd you some spots? I fish from a kayak around the HBC and speak to a couple of LB fisho's from time to time. The usual spots are not what I would call wheelchair friendly as they're across rocks, soft sand etc but I know people catch fish from off the main beaches from time to time. Big Manly, Little Manly, Arkles, Orewa (both ends) are some I've heard have produced some fish recently but not huge amounts. They are reasonably accessible from carparks and boatramps for the wheelchair, although I've not had a lot of experience. You'll need to keep an eye on the tides so you dont have to go too far to reach the water in some spots as they are all fairly shallow beaches.
Regarding Shakepears campground I'm not aware of its facilities, but there is a bit of a walk across uneven ground to the beach there. Maybe call them? I've fished off the eastern end of the beach close in on the yak and had some success although a bit patchy this time of the year. I've seen LB guy's on on that point quite regularly and seen them landing snapper on the odd occasion. Good luck and hope you manage to get out there!