Q: How do I go about filing a Petition to Make Special Patent? Trying to find the applications that are needed to do so

A: The USPTO has a number of different programs.

There is a petition to make special based on applicant's age or health

https://www.uspto.gov/patents-application-process/petitions/23-make-special-age-and-health (Be sure to note the warnings that you need to file in a specific way to avoid having your health information accessible to the public).

At the USPTO Forms page-- https://www.uspto.gov/patent/forms/forms-patent-applications-filed-or-after-september-16-2012

There is a petition to make special under a Cancer Immunotherapy Pilot Program (Form SB/443).

Petition to Make Special based on Age (Form SB/130)

Petition to Make Special under Accelerated Examination Program (Form SB/28)

NOTE -- this is rarely used as the burden of the pre-examination search and Accelerated Examination Support Document is fairly burdensome.

Petition to Make Special Under the Expanded Collaborative Search Pilot Program

The program that I use the most often is not a Petition to Make Special but the Prioritized Patent Examination Program (Track One).

https://www.uspto.gov/patent/initiatives/usptos-prioritized-patent-examination-program

If you qualify as a "micro entity" then the Track One fee is $1000. This will normally get you a first Office Action within 4 months and a completion of examination within a year. The micro entity status is somewhat complicated but you may qualify if your Maximum Qualifying Gross Income is less than $177,117. (this number is adjusted annually).

https://www.uspto.gov/patent/laws-and-regulations/micro-entity-status-gross-income-limit

see form USPTO form SB/15A

Another program I use with some frequency is the Patent Prosecution Highway if I had a positive result in a PCT examination. I suspect that you have not gone down this path but you should learn about this program too.

I hope this helps.

Kevin E Flynn