Hopefully, this doesn't happen, but what happens if the US shutdown continues until the 2020 elections? Does the FEC have funding? Will elections be post-poned? If the FEC does have funding, what would happen if a complete (as opposed to partial) shutdown occured during an election year and said shutdown lasts until the elections?

To address the technical question, FEC will not function during shutdown: https://www.fec.gov/updates/shutdown-announcement-2018/.

Like many federal agencies, the FEC will be unable to provide any services during the government shutdown. Most agency staff will not report to work, and the agency’s offices will be closed to the public. ... You will be unable to contact the FEC during the government shutdown. ...

To address the theoretical question: It's unknown for certain what will happen (and the answers may possibly be impossible to obtain barring a Supreme Court challenge), BUT, philosophically speaking, one thing must be stated:

Elections are run by the states (both Congressional, and Presidential).

FEC facilitates them in certain ways (they enforce rules, mostly related to campaign finance), but FEC is not required to have and hold elections. They were held quite successfully prior to October 15, 1974 formation of FEC.

The Federal Election Commission (FEC) is the independent regulatory agency charged with administering and enforcing the federal campaign finance law. The FEC has jurisdiction over the financing of campaigns for the U.S. House, Senate, Presidency and the Vice Presidency. - from https://www.fec.gov/about/mission-and-history/

Elections are run by individual states and are certified by Congress, the core process wouldn't change. The FEC is a regulatory body that focuses primarily on campaign finances, they aren't really responsible for the core functions of elections. Candidates would still be responsible for following FEC rules, though there may be no auditing/enforcement during elections if they were shut down. The things that would not happen would be the consulting and funding provided by the FEC during elections. Almost all enforcement actions taken by the FEC happen after elections are completed and certified already, not being active would likely further delay enforcement.