The condition of the while loop
while (i <= n && j <= m) {
    if (a[i] > b[j]) {
        c[k] = a[i];
        i++;
    }
    else {
        c[k] = b[j];
        j++;
    }
    k++;
}
is invalid. If for example the array a has n initialized elements then the valid index for the array is in the range [0, n).  Using the expression a[n] results either in accessing memory outside the array a or in reading an uninitialized element of the array.
So you need to write
while (i < n && j < m) {
    if (a[i] > b[j]) {
        c[k] = a[i];
        i++;
    }
    else {
        c[k] = b[j];
        j++;
    }
    k++;
}
Also bear in mind that usually only when an element of the second container (array) is greater than a corresponding element of the first container (array) then it is written in the result container.
That is the if statement should look like
    if ( a[i] < b[j] ) {
        c[k] = b[j];
        j++;
    }
    else {
        c[k] = a[i];
        i++;
    }
if you want to  merge arrays in the descending order.
Or
    if ( b[j] < a[i] ) {
        c[k] = b[j];
        j++;
    }
    else {
        c[k] = a[i];
        i++;
    }
if you want to merge arrays in the ascending order. And you are going to merge your arrays in the ascending order.
These while loops
while (i < n) {
    for (int p = k; p < n; p++) {
        c[p] = a[i];
    }
}
while (j < m) {
    for (int p = k; p < m; p++) {
        c[p] = b[j];
    }
}
can be infinite loops because neither the variable i nor the variable j are being changed within the loops. At least they try to assign the same elements a[i] and b[j] to elements of the array c. And moreover the value of the variable k can be greater than the value of the variable n and greater than the value of the variable m though either array a or the array b has elements that were not yet copied in the array c.
Instead you should write
while (i < n) c[k++] = a[i++];
while (j < m) c[k++] = b[j++];
Pay attention to that you should check that entered values of the variables n and m are not greater than 100.
And there is no great sense to declare the array c with 201 elements instead of 200 elements
int a[100], b[100], c[201];
Pay attention that there is standard algorithm std::merge declared in header <algorithm> that you could use. Here is a demonstration program
#include <iostream>
#include <iterator>
#include <algorithm>
int main()
{
    int a[] = { 1, 3, 5, 7, 9 };
    int b[] = { 0, 2, 4, 6, 8, 10 };
    int c[std::size( a ) + std::size( b )];
    std::merge( a, a + std::size( a ), b, b + std::size( b ), c );
    for (const auto item : c)
    {
        std::cout << item << ' ';
    }
    std::cout << '\n';
}
The program output is
0 1 2 3 4 5 6 7 8 9 10
In the program there are not used standard functions std;:begin and std::end becuase in your program you are using sub-arrays. Though you could write in your program for example
    std::merge( std::begin( a ), std::next( std::begin( a ), n ),
                std::begin( b ), std::next( std::begin( b ), m ),  
                std::begin( c ) );  
Instead of the arrays with predefined sizes you could use standard container std::vector<int> that allows to specify an arbitrary number of elements. With objects of the container you can use the same subscript operator.