How secure is a post doc job offer in an email from the department chair prior to receiving the official written letter?

Don't count on an offer until you have it officially, in writing. A verbal offer is not an offer. An email or call saying you will get the offer is not an offer.

You say you "emailed the postdoc place". Who did you email? If it is an administrator they may not understand the urgency of the situation. I would email the professor you will be working for and (politely) make it completely clear that you have a non-negotiable deadline for replying to another offer. You should also state that if you don't have at least a scanned copy of an official offer from him/her before then, you will have no choice but to accept the teaching position, even though the postdoc is your first choice. If they really want you, they will take the 5 minutes to scan a copy of the letter. If they don't, maybe it's best to turn down that offer anyway.

In the meantime, I would also negotiate with the second place for an extension of the deadline.

An offer in this type of situation is almost always a real offer. However, this kind of delay in getting a written offer is nonetheless quite common and has to do with the way the university's internal bureaucracy works. And as the word "almost" indicates, there is always a chance that something will go wrong, so you are absolutely right to be concerned.

The way it works (in the U.S., and probably in some other countries) is this: postdoc offers are often made by a department chair, but are often subject to approval by a higher university authority such as a dean. Naturally, the dean needs to actually review the file before approving the appointment, and this takes extra time (sometimes several weeks, since deans are busy with many other things) after the department chair or search committee have already recommended making the appointment. In the meantime, the department wants to ensure that they can secure the candidate's commitment to accept the position. Waiting those extra weeks before contacting the candidate is completely impractical and means that they will almost certainly miss out on the opportunity of hiring the person.

The result of this dynamic is that the university will try to play a game whereby the department chair will contact a postdoc candidate with an email, formulated to look as formal as possible (e.g., containing salary and other details, and a response deadline) without actually entering the university into a legally binding commitment. (By the way, IANAL, but just the fact that it's an email rather than a letter is not necessarily the issue; I believe an email could very conceivably be held up in court to be just as binding as a "written" offer, and that it really all depends what the "offer" actually says, including nuances such as whether the word "offer" is used, and whether it contains weasel phrases such as "recommend your appointment to", "subject to approval by", "pending review" etc.)

Note that the way this process is designed is more or less well-intentioned and done in good faith, at least in places I'm familiar with. The goal is simply to achieve an optimal outcome under the constraints of how the university functions, while protecting the university's interests to the extent possible. In all likelihood, no one is trying to scam you into accepting an offer that they will then not offer you. At the same time, as a department chair myself I often think that this way of conducting the university's affairs is somewhat unfair and places an undue burden of worry on the candidate's shoulders. After all, there is always a chance that something will go wrong, there will be a disagreement between the chair and the dean or some other step in the process will fail and the appointment will fall through. And I wonder if when push comes to shove, the university's position of sending emails that are carefully optimized to get people to think that they have a real offer when the language of the email actually avoids making any concrete promises will really hold up in court. At least from an ethical point of view it seems problematic to me. But that's the way things work in many places, and usually things work out in the end.

The bottom line is that a lot of this comes down to a question of trust: did the hiring department manage to instill in you a feeling that they "got you covered" and are backing up their claims with concrete actions? Or are they behaving in a way that seems suspicious and alarming? The lack of willingness to email you a scanned copy certainly seems like a possible red flag (depending on whom you sent it to - never discount the possibility that that person may simply be incompetent or forgetful, so consider looking for other people in the department whom you can turn to for help). Only you can decide how risky the situation feels, and how much risk you are willing to tolerate to secure the position. The only general answer we can give is "an offer is usually a real offer, except when it's not".

This may seem obvious but the fact that you are asking the question implies that you are unsure as to certainty and that alone should encourage you question anything that is not a signed contract.

Personally I would approach this by contacting them directly and explaining your position. If they are 100% on their commitment then expediting the written confirmation you need will not be an issue. A little pressure will work wonders.

Good luck.