If you are out of work because of your employer's not having work for you, and you are able, willing and available to work, you are (if monetarily eligible) qualified for unemployment insurance. If your employer's situation is such that you are working irregular or sporadic hours, either the employer or you can submit the request for partial unemployment detailing the actual hours you have worked in the week to the system and you'll be paid benefits as appropriate.
However, if your employer has work available for you and you cannot do it for a personal reason, such as not having transportation to the job, then they are not going to be on board for your drawing unemployment insurance, which costs them money. Because your not having transportation is a personal issue, something which renders you unavailable for suitable work which has been offered to you by this employer, or for any employer, you are going to be determined to be ineligible for benefits.
If your employer calls you up and offers you work this coming week, and you refuse the work because you do not have transportation, you can bet that if you file a claim for benefits, and are paid, they'll probably complain to the unemployment office. They receive information about who has drawn benefits and for what weeks.
And to file that claim, you'd have to lie on the question about whether or not you are working all hours that are available to you, or whether or not you have refused work this week. So if you received any benefits for that particular week with your car out of service, you did so by committing unemployment fraud, which could very well disqualify you for any future benefits and get you seriously overpaid, which can give you legal problems.
And if your employer said "If you cannot work this week, then you're fired!" and you applied for benefits, then you might not be approved for benefits. If you tell him your car broke down, he can fire you on the spot, or say you basically quit the job by not coming to work. There'd be a decision made about your eligibility for unemployment, and of course there'd be a lot of discussion about whether you're NOW able and available for work, as is, did you get it fixed right away? Have you got transportation now?
Not having transportation is considered a personal issue, not a valid reason to quit a job or fail to show up. And if you were terminated by the employer in this circumstance, it would likely be for not showing up for work for some period, which is very often considered misconduct. Being unavailable for work for any length of time without a valid medical excuse would very likely cause the unemployment system to deny you benefits. Keep that car running or have a back up, keep your baby sitter working or have a back up, pretty much don't miss work for a personal reason that you can control if you want to be approved for unemployment.

