Husk, 8 bytes

!ṁödΣhḊN

Try it online!

Java 8, 220 bytes

import java.util.stream.IntStream;
char a(int n){return IntStream.range(1,n+2).map(i->IntStream.range(1,i).filter(k->i%k==0).sum()).mapToObj(Integer::toString).collect(java.util.stream.Collectors.joining("")).charAt(n);}

Well, it's fast at least. It averages 0.3 seconds to get the 9999/10000th element on my machine. It generates only as many aliquot sums as the index you specified. This means the string will be slightly longer than your index in most cases, since some aliquot sums have 2 or more digits, but for the most part, it only generates as long of a string as we need.

Usage:

public static void main(String[] args) {
    System.out.println(a(0));
    System.out.println(a(4));
    System.out.println(a(12));
    System.out.println(a(9999));
}

Ungolfed:

public static void main(String[] args) {
    System.out.println(all(0));
    System.out.println(all(4));
    System.out.println(all(12));
    System.out.println(all(9999));
}

static int aliquotSum(int n) {
    return IntStream.range(1, n).filter(k -> n % k == 0).sum();
}

static IntStream sums(int n) {
    return IntStream.range(1, n + 2).map(i -> aliquotSum(i));
}

static String arraycat(IntStream a) {
    return a.mapToObj(Integer::toString).collect(java.util.stream.Collectors.joining(""));
}

static char all(int index) {
    return arraycat(sums(index)).charAt(index);
}

Octave, 71 bytes

This one is Octave only. It won't work in MATLAB. A virtual function is created which works on 1-indexed numbers. It can probably be simplified a bit further. Will have a look this evening.

@(x)c((c=num2str(arrayfun(@(n)sum(b(~rem(n,b=(1:n-1)))),1:x)))~=' ')(x)

You can try online here.

Simply run the command above, prefixed with a= or whatever (just so that you can use it multiple times), and then do a(10000) or whatever. It takes about 7 seconds to calculated that the 10000th digit is a 7.

Haskell, 52 bytes

(([1..]>>= n->show$sum[m|m<-[1..n-1],mod n m<1])!!)

Usage example: (([1..]>>= n->show$sum[m|m<-[1..n-1],mod n m<1])!!) 12-> 6.

It's a direct implementation of the definition: foreach n sum it's divisors and convert it into a string. Concatenate all such strings and pick the element at the requested index. Haskell's laziness takes only as much ns from the infinite list [1..] as needed.

Jelly, 13 11 10 bytes

2 bytes thanks to @Adnan and 1 more thanks to @Dennis.

ÆDṖSDµ€Fị@

Try it online!

Uses 1-based indexing. Completes for n = 10000 in under 2 seconds online.

MATL, 16 15 bytes

:"@@q:~fsV]vG)

Indexing is 1-based.

The last test case times out in the online compiler, but it does give the correct result with the offline compiler, in about 15 seconds.

Try it online!

:         % Take input n. Push [1 2 ... n]
"         % For each k in [1 2 ... n]
  @       %   Push k
  @q:     %   Push [1 2 ... k-1]
         %   Modulo. Zero values correspond to divisors
  ~f      %   Indices of zeros. These are the divisors
  s       %   Sum
  V       %   Convert to string
]         % End for each
v         % Concatenate all stack contents vertically
G)        % Take n-th digit. Implicitly display

J, 34 bytes

{[:;1<@":@(-~>:@#.~/.~&.q:)@+i.@>:

This is zero-indexed and uses the formula below to calculate the divisor sums.

Explanation

{[:;1<@":@(-~>:@#.~/.~&.q:)@+i.@>:  Input: n
                                >:  Increment n
                             i.@    Create the range [0, 1, ..., n]
    1                       +       Add one to each to get [1, 2, ..., n+1]
          (               )@        For each value
                        q:            Get the prime factors
                   /.~&.              For each group of equal prime factors
                #.~                     Raise the first to the first power, the second
                                        squared and so on, and sum them
             >:@                        Increment that sum
                      &.q:            Reduce the groups using multiplication
           -~                         Subtract the initial value from that sum
       ":@                            Convert each to a string
     <@                               Box each
 [:;                                Unbox each and concatenate the strings
{                                   Select the character from that string at index n
                                    and return it

PHP, 90 bytes

0 indexed

<?php for(;strlen($s)<=$a=$argv[1];$s.=$n)for($n=0,$j=++$i;--$j;)$i%$j||$n+=$j;echo$s[$a];

Absolutely not subtle or with a clever way of approaching it at all.
Also, as usual, produces three notices which are ignored.

Brachylog, 40 bytes

:2-I,?:?:1yrc:Im.;0.
:1e:2f+.
>.>0,?:.%0

This is 1-indexed, takes about 0.15 seconds for N = 100, 15 seconds for N = 1000. I'm currently running for N = 10000, I'll report the run time once it ends (If my estimation is correct, it should take about 8 hours)

Edit: by fixing premature constraints propagation in Brachylog, it now (on a 3 bytes longer code) takes about 2.5 minutes for 10000 but returns an out of global stack error.

Explanation

• Main Predicate: Input = N

  :2-I,                 I = N - 2
       ?:?:1y           Find the N first valid outputs of predicate 1 with input N
             rc         Reverse and concatenate into a single number
               :Im.     Output is the Ith digit of that number
                   ;    Or (I strictly less than 0)
                    0.  Output is 0
  

• Predicate 1: computes the sum of the divisors

  :1e                   Get a number between N and 1
     :2f                Find all valid outputs of predicate 2 with that number as input
        +.              Output is the sum of those outputs
  

• Predicate 2: unifies output with a divisor of the input

  >.>0,                 Output is a number between Input and 0
       ?:.%0            Input is divisible by Output
  

Python 3.5, 103 93 92 bytes:

R=range;A=lambda f:''.join([str(sum([j for j in R(1,i)if i/j%1==0]))for i in R(1,f+1)])[f-1]

Pretty straightforward implementation of method described in post.

Try It Online! (Ideone)

Does not quite finish within the allotted 5 seconds in the online compiler for input 10000, but it does finish on my machine for the same input within about 8.5 seconds.

Pyth, 26 21 20 15 bytes

@sm`sf!%dTtUdSh

Try it online. Test suite.

Uses 0-based indexing. The program is O(n²) and completes for n = 9999 in about 14 minutes on my 2008 machine.

Mathematica, 51 bytes

Array[##&@@IntegerDigits[Tr@Divisors@#-#]&,#][[#]]&

An unnamed function which takes and returns an integer and uses 1-based indexing. Handles input 10000 instantly.

Explanation

This is a very straight-forward implementation of the definition, making use of the fact that the first n divisor sums are always enough to determine the nth digit. As usual, the reading order of golfed Mathematica is a bit funny though:

Array[...&,#]...&

This generates a list with all the results of applying the unnamed function on the left to all the values i from 1 to n inclusive.

...Tr@Divisors@#-#...

We start by computing the divisors of i, summing them with Tr and subtracting i itself so that it's just the sum of divisors less than i.

...IntegerDigits[...]...

This turns the result into a list of its decimal digits.

##&@@...

And this removes the "list" head, so that all the digit lists are automatically concatenated in the result of Array. For more details on how ## works, see the "Sequences of arguments" section in this post.

...[[#]]

Finally, we select the nth digit from the result.