Inertial navigation is nice but probably has too much drift. So far, the only for individual builders, the only feasible method of dead-reckoning is with map-like 3 IR/sonar beacons where the robot measures the arrival time difference of the IR and sonar signal from each beacon (transmitted simultaneously, but since the IR travels faster than the sonar, there is a time difference). THis is used to calculate how far the robot is from each beacon to form 3 ideally intersecting circles which will tell you where the robot is. More beacons can be used if line of sight is a problem. But the sonar doesn't work well outdoors with wind. THis is the most accurate DIY method available for indoors.
If GPS is not accurate enough, building your own local-area GPS with radio signals is the best way, but for individual users it's really hard to measure the signal travelling times because it is so small (not to mention lots of time synchornization errors between all the transmitters and receivers). A PICAXE is way too slow to handle anything like this, at all. You will NOT be able to pull off the required hardware.
As far as webcams, that's just really hard to process the data to track something (and then even harder to compare it to a map) unless it's an overhead webcam with disctinct colour differences (and the program is probably beyond the reach of a lot of people anyways). If visual landmarks are too hard for you, then probably so is any other method using a webcam. And it's also way too much data for a PICAXE to handle. You will NOT be able to pull off the required software. (well, maybe if it was an overhead camera indoors, but certainly not outdoors with an isosymmetric-view. Either way it would get overl complicated for what you want,
Don't try visual methods or radio-timing methods.
Inertial navigation might be your only method because accelerometers and gyros are fairly cheap and will help your robot stay on course. You might want to have some kind of beacon or something so the robot can regain it's bearings every so often because of the drift involve, perhaps combine inertial navigation with sonar/IR beacons so in the likely even that the robot loses contact with the beacons (wind disrupts sonar reception, IR line of sight interference, etc.). The robot can gauge it's position roughly until it regains contact with the beacons.