Is it common not be able to use tenured department colleagues to write letters of support for tenure?

Your department chair is 100% correct in asking you to remove letters from faculty internal to your university. She is doing you a favor by mentioning this to you!

To be honest, that you think this might be appropriate makes me worry that you may have insufficient familiarity with how the tenure and promotion process works at American universities. The whole purpose of these letters is to get a sense of your academic profile in the academic community at large (i.e., outside of your institution). A good such letter is written by someone with a lot of status in the community and a good perspective...which includes a lack of involvement in local affairs and politics. On the one hand, it is a waste to get a letter from faculty in your department, because faculty in your department already play a much larger role in the tenure and promotion process: they evaluate your entire dossier and vote up or down. On the other hand, if you are asked for five letters of evaluation and you choose two of them from your own department, it will likely create a very strong negative impression: it looks like no one outside of your department has ever heard of you and your work.

I don't mean to be harsh or alarming, but: you need to get a clue pretty fast. Calling attention to the fact that internal letters are not explicitly against the rules of tenure and promotion is absolutely the worst perspective you can have on the situation: in this regard just as is usually the case in life, not everything that is a prohibitively bad idea is explicitly prohibited. I strongly recommend that you approach a trusted mentor (for this inside your department is probably better, although you should take whatever you can get) and say something like "I realize that I have relatively little knowledge of how the promotion and tenure process actually works beyond the formal rules, which I now have reason to believe do not tell the whole story. Can you help me out?"

Good luck.

Yes, totally common. The reason is that these people have an inherent conflict of interest: they would be judge (letter writer) and jury (get to vote on your case) at the same time.

You will want to choose people as letter writers that (i) know you professionally, (ii) are well respected in the field, (iii) have no stake in all of this because they are at unrelated universities.

There are several reasons why even a teaching institution might not want letters from inside your department.

1. It would be redundant, as the department is likely expected to collectively submit a letter.
2. There may be various conflicts of interest that are more likely for departmental colleagues (such as a reluctance to go through a search if tenure is denied and the friendships that naturally develop through working relationships).
3. These letters are partially to give the committee an idea of the candidate's interdepartmental interactions, which are often highly valued at liberal arts schools.

Note that at teaching institutions, as opposed to research institutions, it's not actually uncommon to have colleagues from the candidate's own school write letters or otherwise provide input. Faculty at other institutions may be better/more objective at judging research, but they are much less likely to be in a position to discuss the candidate's teaching and collegiality.

At my own institution, we are required to submit letters from faculty at the institution during some stages of the tenure review. However, there is still a preference that these letters not all come from within the department.