Assuming I have an essential travel need, I would like to know about crossing into tijuana with a receipt + paperwork from the post office that will work as a temporary passport. When you apply for a passport at the post office do they give you something that will serve as a temporary passport? Just was wondering what they give you when you apply for a passport when you never have had one. I am a US citizen.
Passports are issued by the US Department of State, not the US Postal Service. When you file a passport application at a US Post Office, the USPS will issue a receipt for your money payment...but they don't give you a "temporary passport." What you'll get is a receipt, which is not the same as a passport, and no immigration officer from any country will consider it as a valid travel document. The Department of State will later mail your new passport to you.
I'm unfamiliar with the US > Mexico pedestrian crossing at San Ysidro, but I've many times entered Mexico by vehicle without having Mexican Immigration pay any attention to me at all. Because the US has no exit immigration checks, you may be able to walk right across the border into Mexico without issue, although Mexican law may require you to complete an FMM (forma migratoria múltiple) which in practice is rather like a visa or tourist card.
On the other hand, US citizens are required by federal law to enter the US while bearing a US passport. (US citizens must exit the US while bearing a US passport too, but because there are no US exit checks for crossings by land, they may not notice in this case.) While US Customs and Border Protection must admit you to the US even if you have no passport, they will only do so after they're satisfied that you are in fact a US citizen.
Presenting yourself for entry into the US, saying you're a citizen but not carrying a US passport, sets yourself up for a protracted and unpleasant experience while they try to determine your citizenship and identity. At the very least, you will be significantly delayed and the CBP officers will be quite displeased. If they think you're not a US citizen, you could be refused entry back into the US.
In addition to the other good answer, I'll answer specifically for Tijuana.
You say you are crossing by foot into Tijuana - All 3 pedestrian ports of entry into Tijuana from San Diego have passport checks, and you are unlikely to be admitted without a passport. I have heard of one or two people being successful in talking the border guards into allowing them in anyway, however these were people that had already been in Mexico and had only temporarily crossed back into San Diego for shopping (etc). I am aware of far more people who have been denied entry due to no passport.
Visitors to Mexico from the US (and other non-Visa countries) are typically required to have an FMM (visitor/tourism permit) which is obtained at the border, and requires presenting a passport to obtain the FMM, so this would make it difficult anyway.
As noted in the other answer, entering by vehicle usually attracts little attention and no stop, so it is possible to enter Mexico by vehicle without a passport, but I would comment that this isn't necessarily legal, even though it is possible, and if you are stopped by INM (immigration) while in Mexico and you can't provide your FMM or passport you will likely be deported.
I realize this is an old discussion but I just spoke with a relative who walked across for dental work at Los Palomas, which is south of Columbus New Mexico. No passport. He did go prepared though. First, he called the main CBP office in El Paso. It was explained to him that for an American Citizen with sufficient identification the border guards must allow you back in.
In my relative's example, he carried a photocopy of his original birth certificate and of course his New Mexico driver license. The CBP guard when he crossed back with obvious swollen mouth and some bleeding gums from dental work still hassled him. But he simply said clearly and politely as his mouth situation permitted that he knew the law, he'd shown proof of American birth and photo ID and he could not be refused.
Took maybe two minutes in that case, but back right back into the USA he walked.