Of the writers I'm familiar with, Hugh Kennedy seems to go into this type of practical military question the most, that being said, as he notes in a caveat in The Great Arab Conquests:

There is no reliably dated archaeological evidence of Arab military equipment from this period, no surviving swords or armour. Instead we have to rely on incidental mentions in narratives and poetry which, except in exceptional circumstances, rarely provide detailed descriptions.

I would also add that in many instances these narratives are of spurious provenance and might well be from much later. We also have to contend with the fact that it was extremely common in the late antique/early medieval period to fill in the gaps of historical knowledge when writing at a later date by imposing facts from the present onto the past. So much so that in some instances chroniclers ostensibly writing about a previous century might actually be more valuable for their own time period.

That being said, in the early conquest armies, again all of this from Conquests:

There was no question of any uniform: each man would dress in what he could find, and what he could afford.

As a result:

Military equipment was one of the most sought-after items of booty when an army was defeated or a city taken. A lively market in weapons and armour frequently ensued.

As to the specific equipment, however:

Chain-mail body armour was worn, although the number of men who could afford it must have been quite small: in 704 it was said that in the whole vast province of Khurasan there were only 350 suits of mail for about 50,000 warriors. Coats ofmail were handed down from one generation to another, and new ones, brilliant and shiny, were extremely valuable. Head protection came in two forms. There was the mighfar, known in the history of Western armour as an aventail. This was essentially a hood of chain mail which was extended down at the back to protect the neck. Alternatively, there was a rounded helmet known as a bayda or egg. A fully equipped warrior must have been quite well protected, at least as much as the Norman warriors of the Bayeux tapestry, but most of the rank and file must have been much less fortunate, fighting in cloaks and turbans which would have left them very vulnerable.

Obviously this is written significantly after Muhammad's own liftime, so a comparative lack of formal armor and equipment can plausibly be extended backwards.

As he elaborates slightly in The Armies of the Caliphs:

Mail was certainly known in the Roman army from the third century6 and by the time of the Muslim conquests it was probably the main form of body armour for 􏰀􏰋􏰁both Byzantine and Sasanian soliders. In the Qur'an, the making of coats of mail is one of the blessings conferred on God on David. The early Islamic sources treat the coat of mail as a standard piece of military equipment. It could be worn under a cloak....There are also references to the practice of wearing two coats of mail (dir’ayn), the under one being shorter or even made of fabric or leather.

Mail required a considerable investment and it is likely that only a small proportion of the soldiers in any army could afford it.

It's perhaps worth noting that there's every reason to believe that various forms of leather armor would have been readily available or at least that leather itself was in good supply. As Patricia Crone notes in her monograph Meccan Trade and the Rise of Islam it seems that leather trading was perhaps one of the more plausible economic activities of the pre-Islamic era. That materials like leather were used might a long way to explaining why we have no such archaeological evidence.

Also, even lacking body armor, the early Arab armies would also have had recourse to shields, as Kennedy notes in Armies:

The Syrian forces at Siffin were lined up with spears (rimah)􏰾 and shields (daraq) and helmets 􏰼􏰀(bayd)􏱁􏰾 on their heads.

The helmets were described by Abu Ubayda, one of the companions of the prophet, as quoted in the chronicle of Tabari as:

'composed of plates like the bones of the skull, the edges whereof are joined together with nails; and sometimes of one piece’

So as a TL;DR and per Kennedy's conclusion:

The armor quality of early Muslim forces would have varied considerably, but at the elite or wealthy end of the spectrum chainmail body armor comparable to that used by Byzantine or Sassanian forces was available and well-known. For the common soldier, shields, basic helmets, and in all likelihood various forms of leather or informal armor would have been common with a large portion of the troops being essentially un-armored.