Are there any disadvantages related to purchasing the property in the name of my revocable living trust? 

From what you described above I am not sure why you'll need the consent of this person who no longer owns the land, but there really isn't enough information in what you have provided to help much. 

My actual question is, once a person has submitted an application request through Apartments.com, is that giving consent to call their references/landlords/employers to verify the data or do you need to get more consent?

I just got a reply back from her when I asked her to provide me with consent to verify her employment and she said "you have the wrong number"  There was just too much that was fishy...and I'm so glad that BP was here and that I took the extra day to wait until I could contact her employers HR department.

Normally it is the owner of the property that needs to make a case to the board but it does specify that it may be someone else with interest in the property with the owners consent.

@Kevin Hofstee - I don't know the in's and out's, but yes my personal residence is in a revocable trust and I was able to refinance it with personal financing in my name. 

For example, what you describe sounds illegal in California without the consent of both you and your prospective tenant. 

Hi Darius, It says "borrower shall occupy the property as principal residence for at least one year after the date of occupancy, unless Lender otherwise agrees in writing, which consent shall not be unreasonably withheld, or unless extenuating circumstances exist which are beyond borrower's control."

I know my parents would never give me consent to sign the contracts, or sign them for me, which is sad. 

The other Person signed onto the lease without my knowledge or consent.