Wolfram Language (Mathematica), 50 47 42 25 bytes

{}⋃Select[x/.Solve[#~FromDigits~x==0],IntegerQ]&

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Update: using Luis Mendo's fact, golfed off another 3 bytes

Pick[r=Range[s=-Tr@Abs@#,-s],#~FromDigits~r,0]&

Getting sloppier with the bounds, we can reduce this 5 more bytes per @Not a tree's suggestion:

Pick[r=Range[s=-#.#,-s],#~FromDigits~r,0]&

After posting this, OP commented allowing "native polynomials", so here's a 25 byte solution that accepts the polynomial as input. This works because by default Mathematica factors polynomials over the integers, and any rational roots show up in a form like m*x+b that fails the pattern match.

Cases[Factor@#,b_+x:>-b]&

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Wolfram Language (Mathematica), 50 47 42 25 27 bytes

{}⋃Select[x/.Solve[#~FromDigits~x==0],IntegerQ]&

Try it online!

Update: using Luis Mendo's fact, golfed off another 3 bytes

Pick[r=Range[s=-Tr@Abs@#,-s],#~FromDigits~r,0]&

Getting sloppier with the bounds, we can reduce this 5 more bytes per @Not a tree's suggestion:

Pick[r=Range[s=-#.#,-s],#~FromDigits~r,0]&

After posting this, OP commented allowing "native polynomials", so here's a 25 byte solution that accepts the polynomial as input. This works because by default Mathematica factors polynomials over the integers, and any rational roots show up in a form like m*x+b that fails the pattern match.

Cases[Factor@#,b_+x:>-b]&

As @alephalpha pointed out this will fail for the case where zero is a root, so to fix that we can use the Optional symbol :

Cases[Factor@#,b_:0+x:>-b]&

This parses fine Mathematica 11.0.1 but fails and requires an extra set of parentheses around b_:0 in version 11.2. This takes up back up to 27 bytes, plus two more after version 11.0.1. It looks like a "fix" was put in here

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Wolfram Language (Mathematica), 50 47 42 25 bytes

{}⋃Select[x/.Solve[#~FromDigits~x==0],IntegerQ]&

Try it online!

Update: using Luis Mendo's fact, golfed off another 3 bytes

Pick[r=Range[s=-Tr@Abs@#,-s],#~FromDigits~r,0]&

Getting sloppier with the bounds, we can reduce this 5 more bytes per @Not a tree's suggestion:

Pick[r=Range[s=-#.#,-s],#~FromDigits~r,0]&

After posting this, OP commented allowing "native polynomials", so here's a 25 byte solution that accepts the polynomial as input. This works because by default Mathematica factors polynomials over the reals, and any rational roots show up in a form like m*x+b that fails the pattern match.

Cases[Factor@#,b_+x:>-b]&

Try it online!

Wolfram Language (Mathematica), 50 47 42 25 bytes

{}⋃Select[x/.Solve[#~FromDigits~x==0],IntegerQ]&

Try it online!

Update: using Luis Mendo's fact, golfed off another 3 bytes

Pick[r=Range[s=-Tr@Abs@#,-s],#~FromDigits~r,0]&

Getting sloppier with the bounds, we can reduce this 5 more bytes per @Not a tree's suggestion:

Pick[r=Range[s=-#.#,-s],#~FromDigits~r,0]&

After posting this, OP commented allowing "native polynomials", so here's a 25 byte solution that accepts the polynomial as input. This works because by default Mathematica factors polynomials over the integers, and any rational roots show up in a form like m*x+b that fails the pattern match.

Cases[Factor@#,b_+x:>-b]&

Try it online!

Wolfram Language (Mathematica), 50 47 42 25 bytes

{}⋃Select[x/.Solve[#~FromDigits~x==0],IntegerQ]&

Try it online!

Update: using Luis Mendo's fact, golfed off another 3 bytes

Pick[r=Range[s=-Tr@Abs@#,-s],#~FromDigits~r,0]&

Getting sloppier with the bounds, we can reduce this 5 more bytes per @Not a tree's suggestion:

Pick[r=Range[s=-#.#,-s],#~FromDigits~r,0]&

After posting this, OP commented allowing "native polynomials", so here's a 25 byte solution that accepts the polynomial as input.

Cases[Factor@#,b_+x:>-b]&

Try it online!

Wolfram Language (Mathematica), 50 47 42 25 bytes

{}⋃Select[x/.Solve[#~FromDigits~x==0],IntegerQ]&

Try it online!

Update: using Luis Mendo's fact, golfed off another 3 bytes

Pick[r=Range[s=-Tr@Abs@#,-s],#~FromDigits~r,0]&

Getting sloppier with the bounds, we can reduce this 5 more bytes per @Not a tree's suggestion:

Pick[r=Range[s=-#.#,-s],#~FromDigits~r,0]&

After posting this, OP commented allowing "native polynomials", so here's a 25 byte solution that accepts the polynomial as input. This works because by default Mathematica factors polynomials over the reals, and any rational roots show up in a form like m*x+b that fails the pattern match.

Cases[Factor@#,b_+x:>-b]&

Try it online!