How does a registrar such as GoDaddy or Namecheap acquire their own domains (e.g.
https://www.godaddy.com/)? Do they just use their own infrastructure, or do they have to pull some strings related to their status as registrars?
Short answer: a registrar can register their own domains just like anyone else.
Behind the domain name system is a network of registry operators; in the case of gTLDs (global top level domains - eg. com, net, org and many, many others) the registry operator is ICANN.
Each registrar (GoDaddy, Namecheap etc.) is accredited by the registry operator to register domain names at wholesale rates on behalf of their clients. Registrars then provide their own services on top of this - which can range from a simple 'registration service' to 'decent customer service' to full blown clustered web hosting solutions.
When a registrar wants to register their own domain name, they can do it just like anyone else - by doing it through a registrar. They could technically choose any registrar, but obviously, they'd choose themselves because it'd cost less (they have access to the wholesale rates), and it'd look pretty silly to have it registered somewhere else anyway. :)
You can see where a domain is registered by doing a whois.