"Will I need.."
Yes
The difference between one place and another will likely only be a few quid; not enough to be worth the hours endlessly scouring the internet. My last ones were from Amazon, maybe 13 quid for the pair. Changed the cold, 6 months later the hot was leaking, glad I'd already bought it. If it'd been a customer job I'd have likely done both
It's not a difficult job; there will likely either be a tiny grub screw, probably where you can't see at the bottom of the tap lever, that holds the lever on, or capnonnthebsidenof the lever that prises off to reveal a screw. If it's a grub screw, getting a small Allen key to fit it is possibly the bigger challenge/not something you might have in the tool kit.. once that's undone just enough to allow the lever to come off you'll be looking at a brass coloured lump part of which is a big nut. Undo it, holding the tap for extra bracing (leaning on the spanner anticlockwise will want to bend the whole tap towards you, rocker some small counterforce away from you on the centre stem to the small bolts that hold the tap to the worktop aren't taking all the force)
Fit the new cartridge and nip it up with the spanner; it doesn't need to be tightened to oblivion, as a rubber seal provides the water tightness, not the tightness of the nut
Refit the lever, remembering to make sure the valve is in the closed position before mounting the lever vertically. If you can't get the lever perfectly vertical enough for your OCD because one spline is too much and the adjacent one is not enough, nip the nut up a tiny bit more