WASHINGTON—Hundreds of thousands of veterans face yearslong delays in their appeals of disability rulings because of a backlog of cases choking the Department of Veterans Affairs, but a new round of efforts aims to improve and speed up the claims-and-appeals process.

This backlog causes a number of problems, according to the department’s inspector general and veterans advocates, as well as VA statistics. Rushed rulings on initial claims can be riddled with errors. Veterans who appeal their cases typically wait between three...