Yes to both questions. Federal contractors are responsible for ensuring that their third party vendors are collecting the self-ID data for race, gender, veterans, and individuals with disabilities. Likewise, it is incumbent on the federal contractor to solicit this data for all candidates who meet the minimum qualifications. Having them apply to the website is a good way to accomplish this.
You can use this OFCCP audit checklist to ensure you're doing what is required to maintain OFCCP's regulations including VEVRAA, Section 503, and EO 11246. Or request a demo to streamline your compliance and recruiting efforts.