Dear Rabbi, In a situation where one is not able to put on tefillin in the morning because of an upset stomach, should one at least put them on for shema and amida or is it best to not put them on at all?

Shalom, Thank you for your question. May I first send you blessings for a speedy return to good health, and to overcome all stomach problems. You are correct that one needs to be careful when putting on tefillen to make sure that they can keep their body clean – which includes not passing wind, and certainly not having to run to the bathroom, with tefillen on. The answer to your question depends on what you mean by "an upset stomach". If you are so sick that you feel a need to vomit, or feel that you have to constantly run to the bathroom, then it is improper to pray, let alone put on tefillen. If, however, you problem is that you need to pass wind a lot, then the law is different. If you estimate that you can control your flatulency for at least a short time, you should put tefillen on between the blessing of "Ahavat Raba" and "Shema", (with the blessings) and then remove them directly after Shema. Of course, if you can have them on for longer with no doubt of passing wind, you should leave them on for the Amidah also. If even this is not possible, then if during the course of the day you see that your stomach has calmed down, you can (and must) put tefillen on – even if it is the afternoon. In such a case you should recite the Shema with tefillen on. Blessings.