While having internet connectivity on board your mobile home would seem like the ultimate freedom option, there are a few alternatives especially if you are only checking emails, making bookings and updating your social media accounts.
Internet cafes, library networks, buses and commercial retailers (ie McDonalds) may provide free or paid access to the internet through their server/ modem via wi-fi.
I'm not a fan of these options as I have concerns about security on public access terminals.
However how they work is that you buy an allocated number of minutes or amount of data, log in to the providers webpage to enter the access code, and it's all go.
Some providers are nation-wide so any pre-paid unused minutes/data, after you have logged off, can be elsewhere on your travels.
For free or free-to-customer access, try the Aotearoa People's Network member public libraries (not all got to join before the scheme closed), McDonalds, Starbucks, Burger King, i-Site Visitor Information Centres, Warehouse Stationery or your favourite cafe or restaurant. Auckland have their own service to the public. You can even get it while crossing the Cook Strait on the Bluebridge Ferry!
These are less likely than internet cafes to have security issues although not being in the know about what measures they have taken there is no guarantee. [It would be helpful to find out how they prevent cross transmission between devices, how tamper-proof their cable is, and what if any association they have with the GCSB!]