The fastest vaccine developed is a mumps vaccine, which took 4 years. Where is the hope that we will find a COVID vaccine in less time?

Expectations for accelerated vaccine development are based on two main points.

1. Groups have been working on vaccines for SARS-CoV-1 since 2003, and much of that work is applicable to SARS-CoV-2 (the COVID-19 virus). Here is a review article discussing work on SARS-2 vaccines based on work on SARS and MERS. One example, transgenic mice are available that express the human version of ACE2, the host receptor protein that allows SARS coronaviruses to enter a cell. These mice allow vaccine testing that is much more informative than cell culture tests, and weren't developed until 2006-2007. SARS was eliminated in 2004, so we can estimate it would take at least two years to develop an animal model for COVID if it wasn't already available.

Here is a diagram from the linked review showing timelines for traditional vaccine development and the accelerated SARS-CoV-2 vaccine development:

2. Several pharma companies have developed "plug and play" vaccine platforms which allow rapid manufacture of vaccine candidates once a molecular target is known. In this case, the target is the 'S' glycoprotein. Once the sequence that encodes for this target is known, it can be inserted into the platform and produced quickly.

Here is another diagram from the same review describing some of these platforms:

EDIT - I wanted to emphasize a third point: If you notice the "Production (at risk)" text in the accelerated development diagram, it means that pharma companies are producing large amounts of their vaccine candidate before testing is complete, and therefore running the risk that their warehouses full of product will be rendered worthless by bad outcomes in the clinical trials. Moderna had started production of their mRNA vaccine candidate by July 15 2020, so this gives them a several month head start if the trials turn out well.