Were there loads of siddurim (prayer books) in the time of the tosaphot?

No.

תוספות מסכת תמורה דף יד עמוד ב
דברים שבכתב אי אתה רשאי לאומרם בעל פה - וא"ת היכי קאמרינן מזמורים על פה וי"ל דאין להקפיד רק מה שכתוב בחומש אמנם קשה היכי קרינן ויושע וקריאת שמע וי"ל דאין להקפיד אלא בדבר שמוציא אחרים ידי חובתן

Commenting on the Talmudic law that Bible may not be said by heart, the tosaphot asks why "we" say psalms by heart (as you can soon see, from context it is talking about the psalms that are in the daily prayers, like the pesukei de-zimra). If the answer is that the Talmud's rule really only applies to Torah, then it is necessary to explain how in the daily prayer service "we" read additional parts of the Bible from the Torah, the Song of the Sea in Exodus and the Shema, from Deuteronomy, by heart. (The tosaphot's answer is not relevant to this discussion, but I posted it anyway).

This tosaphot seems to incidentally present proof that in its time and place prayers were mainly recited by heart and, presumably, siddurim (prayer books) were scarce.

Conversely, it might not be taken as proof for that assertion since when the tosaphot asks היכי קאמרינן מזמורים על פה it may not mean that the usual practice was to recite by heart. Maybe it meant occasionally people look away from the siddur, just as today many, or most, say aleinu by heart. We certainly have siddurim. In other words, perhaps it is asking how we can justify reciting the psalms in pesukei de-zimra by heart, as people are sometimes wont to do, if one looks outside the siddur.

I think that's the weaker interpretation, for a couple of reasons.

1. Who said that's justifiable? If everyone has a siddur and someone chooses or happens to say pesukei de-zimra by heart, but could read it, why is it a given that this is acceptable? [1]

2. Books had to be hand copied then, and we know that they were not cheap. I just did a quick check of how many words the daily shacharit alone is, and although I used a contemporary nussach and the text included things like multiple repetitions of kaddish, I counted over 12,000 words. Even if we reduce the number to 10,000, that is not such a short text. Producing one of those hand-written--and, mind you, this is only the text of shacharit, and not many other daily prayers, it doesn't include hallel, etc. for each and every person is unlikely.

[1] Alternatively this could be a type of thinking formerly found in rabbinic literature (although occasionally still), where popular practice was assumed to be proper, rather than improper. However, I don't think this is the case here because these kinds of justifications didn't pertain to the occasional, individual practice, but rather to things which masses of people did which would seem to contravene halakhah, hence the assumption that it didn't so one only had to try to understand why not.