Is there a lack of historical evidence for the origins of Islam? And what does this mean? (specifically, the 7th century is said to have a "blackout" of information)

Tom Holland is a decent writer, but he is not a specialist historian working on this period and exaggerated the difficulty of working in this field. It is true that we don't have many sources for most of the seventh century, but there definitely wasn't a 'blackout' of information. Here are the basic facts:

All Islamic histories were written at least 150 years after Muhammad, before then traditions were passed down orally. Even these early histories can be a bit dodgy, as they were useful for legal reasons and some, like Ibn Ishaq (who wrote the first biography of Muhammad), were criticised by their contemporaries for making mistakes in their narratives.
The Byzantine historical tradition, which provides us with the bulk of the information on the Middle East, also died out in the early-seventh century. The last Greek ecclesiastical history ended with Evagrius in 593, Theophylact Simocatta's secular history ended in 602 and the last chronicle, the Paschal Chronicle, ended in 630.

So yes, we don't have any obvious contemporary histories from this period, which is pretty bad when compared to the sixth century, which was covered by a succession of Greek ecclesiastical/secular historians. However, even if we ignore all Islamic traditions (and I don't think we should, because I think some facts can still be recovered), there are still some references to Muhammad and early Islam in non-Greek sources, particularly in Syriac and Armenian. The best work on this is done by Robert Hoyland, who collected basically all mentions of early Islam in his book Islam as Others Saw It, and the rest of my answer will be based mostly on his book/articles. Based on the evidence he collected, he concluded that the general narrative of early Islam can be confirmed by its contemporaries, though obviously the finer points are still debatable (and rightfully so).

There are two possible early mentions of Muhammad and his movement. The first is an anti-Jewish pamphlet (supposedly) written in Africa in 634, which claimed that some Jews were convinced by a new unnamed prophet amongst the Saracens (Arabs). This suggests that it was a religious movement the Byzantines were worried about and not just an invading army. The second is a short note written on a Syriac Gospel book around 636, presumably by a monk writing down the momentous events he just experienced - the most important line being:

... many villages were ruined with killing by [the Arabs of] Muhammad and a great number of people were killed and captives [were taken] from Galilee as far as Beth

This note however is faded and the reading uncertain, so Hoyland prefers a chronicler's account written around 640 as the 'first' mention of Muhammad, which said that:

In the year 945, indiction 7, on Friday 4 February (634) at the ninth hour, there was a battle between the Romans and the Arabs of Muhammad

Based on these three early examples, I think it is clear that Christians recognised the importance of Muhammad for an Arabic religious movement that suddenly attacked the Byzantine Empire. Of course, there is also evidence that early Islamic beliefs were different to later ones, but the academic consensus right now is that general gist of the religion was in place already, as the Quran is also generally considered to be an authentic document from this period. We can argue about whether early Islam was more closely affiliated with Judaism or Abrahamic faiths in general, but it would be wrong to say that Islam didn't exist in the seventh century.

The last source I'd like to talk about is the Armenian History of pseudo-Sebeos written in the 660s. He wrote extensively on the rise of Islam and one particular passage is I believe the first mention of early Islamic teachings, a passage that meshes quite well with what we know from later Islamic traditions. Again, it's worth quoting in full:

At that time a certain man from among those same sons of Ishmael whose name was Mahmet, a merchant, as if by God’s command appeared to them as a preacher [and] the path of truth. He taught them to recognize the God of Abraham, especially because he was learned and informed in the history of Moses. Now because the command was from on high, at a single order they all came together in unity of religion. Abandoning their vain cults, they turned to the living God who had appeared to their father Abraham. So Mahmet legislated for them: not to eat carrion, not to drink wine, not to speak falsely, and not to engage in fornication.

He said: ‘With an oath God promised this land to Abraham and his seed after him for ever. And he brought about as he promised during that time while he loved Israel. But now you are the sons of Abraham, and God is accomplishing his promise to Abraham and his seed for you. Love sincerely only the God of Abraham, and go and seize your land which God gave to your father Abraham. No one will be able to resist you in battle, because God is with you.'

This looks pretty similar to what we know about Islam right? So clearly some religious beliefs had solidified by the 660s. At the same time, we have Mu'awiyah, first caliph of the Umayyad Caliphate, who, as the documentary explained, didn't mention Muhammad at all on his coins or buildings. Yet his contemporary pseudo-Sebeos clearly thought Muhammad was a big deal in Islam. The caliphate had adopted the administrative practices of the Byzantine/Persian empires, so in the first few chaotic decades of Arab rule coins continued to follow Byzantine/Persian models and featured images of emperors, Christian crosses and Zoroastrian symbols. Pure necessity, rather than because Mu'awiya was not a Muslim, is I think a better explanation for why there were no mentions of Muhammad during Mu'awiya's reign. Moreover, pseudo-Sebeos confirmed the later story of Muhammad being a merchant who began his religious mission out of nowhere, so I don't think it would be fruitful to dismiss all later sources for early Islam either. After all, some outlandish stories, such as how Arabs took a city in Iran via an underground tunnel, can occasionally be backed up by contemporary sources, in this case by the Khuzistan Chronicle.

So whilst we don't have a lot of contemporary information about this period, we can still produce a history of early Islam by piecing together all the pieces. Holland's approach is certainly not the mainstream view! I would highly recommend Robert Hoyland's book if you want to look at the evidence yourself, as well as James Howard-Johnston's Witnesses to a World Crisis, which presents a detailed analysis of all the sources available.

Let me know if you have any questions :)