Observations begin with year 0.
Let $c_{i,j}$ the number of comets sighted the first time on year $j$ that have an $i$ years orbit, with $i\in[1,100]$ and $j\in [0,i-1]$.
Let $n_k$ be the number of comets sighted on year $k$.
Each year of observation gives an equation:
$$\sum_{i=1}^{100}c_{i,k\bmod i}=n_k$$
The total number of comets is $$\sum_{i=1}^{100}\sum_{j=0}^{i-1}c_{i,j}$$
There are $100\times 101/2=5050\; $ variables, so one could think that this number of years of observation would give enough equations. But it is likely that many equations will turn out to be linear conbinations of others. On the other hand, this will be a system of linear Diophantine equations (i.e. using only integers). In this case, resolution may not require as many equations as there are variables.
To be sure to observe everything, it is better to do it until the set
of comets return to its initial configuration, which is after a number of years that is the least common multiple of the first hundred integers: I have not computed it, but it seems to be somewhat longer than the current age of the universe (probably even when measured in seconds - it broke my LCM calculator). But as Tycho-Brahe taught us, astronomy is a science for the patient.
Maybe a better analysis would show that less years may be needed (I
doubt it, but I am not sure). Else, we just get an
approximation, or improve the precision of the solution as time goes by and more observations are available. Anyway, we know that there is some indeterminacy in the
problem that prevents distinguishing an obit from its harmonics, as remarked in another answer. So we
know we will not get enough linearly independent equations anyway.
Well, once we have the equations, we solve them as much as possible,
and get a set of possible answers, depending on a few remaining
variables that can be freely chosen.
Then we can impose a further constraint on the total number, which reduces by one the degree of freedom on the answer.
This is probably not be a complete resolution of the problem. Only integer solutions are
acceptable, and it is not clear to me that resolving the equations give
only integer answers.
Systems of linear equations are normally solved over a field, and
integers do not form a field. So some of the solutions found will be
rational numbers, as rationals do form a field. But that is not
acceptable. Some integer solutions must exist, though, as it is an
hypothesis of the question: parameters of the problem result from
observation of a solution (except for the total number of comets which can only be given by an oracle). So integer solutions will have to be
extracted from the set of rational solutions, I suppose with
divisibility considerations.
But my memory of the properties of linear
diophantines systems and their resolution is gone, whatever it was at an earlier age.
So I leave this for someone else, unless I find enough information on the web, and the time to read it.
This calls for a few remarks.
His strange behavior made me react. I am wary of technical questions
supposedly naive, because the asker is often gone when I post the
answer, not to come back ... and I hate it. Then the reaction
of the OP gave me the clue: he knew. That was motivating.
I have an excuse: I am not a native speaker. What is the meaning of HCBP?