If all convicts are unable to rent, the homeless population would be much higher than it already is. There is no law that prohibit landlords from renting to someone convicted of relatively minor criminal law offenses.
In fact, there is no law that would prohibit a landlord from renting to a tenant with any type of criminal law history. In cases such as tenants who are registered sex offenders, public pressure often discourages landlords from renting to prospective tenants with certain types of criminal law histories.
On another hand, discrimination based on criminal law convictions is not unlawful. So, there may be some landlords who would not rent to anyone with any type of criminal law histories.
WA allows certain types of convictions to be vacated. However, this is a once-in-a-lifetime chance. Using the chance on a relatively minor conviction may not be its best use.
Depending on what actually happened in your court case, you may not actually have a conviction. The charge against you may have been dismissed is that was your agreement with the prosecutor.
Even if you do have a conviction, you likely are not eligible to have your conviction vacated for at least another year.
What you should do is review the specific facts with your attorney to find out what legal options you may have.
It depends on whether or not the landlord runs your criminal history and does a credit check. If they don't, then it probably won't matter. However, if they do, then it may not reflect very good upon who you are to the landlord and what they think about you. There are further consequences that go beyond jail time or fines.
There are ways to get it off of your criminal record. Call an experienced attorney about getting it vacated.