If you want to deliver groceries,you might want to check out SYSCO,or Ben E Keith,or US Food Service. Those are local jobs. Of course you have your local LTL freight companies,but like you said,it can be tough getting on those. The major carriers that offer local jobs are usually in the larger cities. I dont know of any websites right off hand that have local jobs. You may want to check out
www.careerbuilder.com or
www.monster.com and see what might come up.
www.fleetdirectory.com has a list of trucking companies. Not all companies are on that site,but they do have a large list.