If any Realtor actually told you that you "should move to Englewood or Garfield Park," or to any other neighborhood, for that matter, they're breaking the law prohibiting steering, that is, channeling people to particular neighborhoods based on their perception of "where they belong."
It's a crime and you should report it to the local Realtor association, in this case, the Chicago Association of Realtors. You also can contact the Illinois Department of Financial and Professional Regulation (IDFPR), the state agency that issues real estate licenses. The IDFPR disciplines hundreds of licensees each year for violations of the state statute.
Regarding your question, I think many landlords might have more of a problem with your pets than with your voucher. As demand for rental housing in the city has increased dramatically over the past several years, so have rents and competition for vacant apartments.
Landlords, given a choice, most often will rent an apartment to a tenant without pets. Many landlords will not consider any applicant who has a pet, and explicitly state in the lease that no tenant may keep pets.
Why? Because there's a perception on the part of landlords (sometimes validated by experience) that tenants oftentimes don't control their pets, allowing them to damage the property and annoy neighbors. You'll need to find a landlord who likes animals and is pet-friendly.
Your housing voucher is another matter entirely. My colleague stated below that the housing voucher sets you in a protected class for the purposes of tenant screening. In Chicago, local ordinance states that "source of income" cannot be used to discriminate, as long as the source of that income is legal. Therefore, your housing subsidy must be considered just as any other income from work, Social Security, or pension benefits.
Under this standard, a person who earns, say $2000 monthly from employment is on the same footing as a person with a $700 housing voucher and $1300 in Social Security benefits. A landlord could still consider creditworthiness based on record of payment, but with regard to income they are essentially the same. Most landlords want tenants to keep housing expenses to no more than about 30% of gross (pre-tax) income.
One continuing issue with the Housing Voucher program is the inspection requirement. Many landlords are not willing to make any repairs the housing authority requires after inspection. Although the letter of the law requires landlords to accept your application without regard to Housing Voucher status, if they have another equally-qualified applicant who is not requiring an inspection or repairs, they likely will choose that applicant, since it potentially costs them less money and less risk.
There are landlords who understand the Housing Voucher program and welcome tenants who have a voucher. I agree with my colleague below who suggested that you ought to work with a Realtor who specializes in tenant representation.
Make sure the person you hire is licensed; they should show you their "pocket card" at your request, which is a copy of their state license. Beware of anyone who refuses to provide proof that they're licensed or says that they "work under" someone else's license -- this is illegal. Also beware of anyone who wants to charge you a fee upfront for service -- in almost all cases, it's the property owner who pays to have an apartment rented.
The only fees that tenants commonly pay are a nominal application fee (usually less than $25) and credit check/background check fee (also in the range of $25). These fees can vary from company to company, landlord to landlord, but should not be exorbitant.
The Housing Voucher program can be complicated and frustrating, but if you're patient (and get the help of a Realtor) you can find a good landlord in the area you wish to live. Good luck to you in your search!
Don Pasek, CIPS, TRC, ADPR
Omniterra Real Properties
I am in total agreement with you.
Unfortunately the low vacancy rate due to people being nervous about purchasing has resulted in a landlords market. One of the things you can do is walk the areas you would like to live in and see if you see landlord for rent signs. Make an appointment with the landlord to view the apartment and go from there.
Of course you can also work with brokers and leasing agents familiar with the voucher program. I would recommend doing both.
Feel free to contact me should you need any assistance.