Does there exist a public database of RSA-2048 moduli and their factors? Provided some modulus for RSA 2048, is there a way I could look up if


the modulus 
and their factors


exists online?
TL;DR
It's not possible, because there are way too many prime numbers.
It's possible that there might be a database with some of these numbers, but not a database with all the possible numbers. There are just way too many prime numbers to fit them all in one database.

How many 1024-bit prime numbers are there?
To get a 2048-bit modulus you multiply two 1024-bit prime numbers together. Using $$\frac{n}{\ln(n)}$$ we can estimate the amount of prime numbers with the length of 1024-bits. The amount of 1024-bit numbers is $2^{1023} - 1$.
$$2^{1023} - 1 \approx 8.9 \times 10^{307}$$
We then apply the approximation:
$$\frac{8.9 \times 10^{307}}{\ln(8.9 \times 10^{307})} \approx 1.25 \times 10^{305}$$
We now have an approximation of how many 1024-bit prime numbers there are.

How large is the database?
If you want to put them all in a database you just have to calculate the number of these primes multiplied with their size (which is of course 1024-bits). That gives you the size of your database in bits:
$$1024 \times 1.25 \times 10^{305} \approx 1.28 \times 10^{308}$$
So as you can see this database would be way too large and it wouldn't be practical to even try to fit them in any database.