Can I start on my HRAP packet if I only have projections?

You should be able to apply if you don't have specific assignment instructions yet. The process to apply is not difficult and the regulation spells out exactly what needs to go onto the DA 31. Be alert to the caveats of this program, such as performing HRAP in a geographical area you are familiar with, no more than 50 miles from a recruiting station, etc. If you haven't reviewed the reg, see para 7-3, AR 601-2.
https://armypubs.army.mil/epubs/DR_pubs/DR_a/pdf/web/ARN21980_R601_2_FINAL.pdf