Can I use ListDeconvolve or any other functions to get the original kernel?

In this case, the following works just fine:

LeastSquares[
 Transpose@Through[{RotateLeft, Identity, RotateRight}[data1]],
 conv
 ]

{0.333333, 0.333333, 0.333333}

More general with kernel length 2 k + 1:

a = RandomReal[{-1, 1}, 100];
k = 20;
ker = RandomReal[{-1, 1}, 2 k + 1];
b = ListConvolve[ker, a, k + 1];


ker1 = LeastSquares[
   Transpose[RotateRight[a, #] & /@ Range[-k, k]],
   b
   ];

Max[Abs[ker - ker1]]

2.27596*10^-15

This is a bit roundabout but one can treat this as a polynomial algebra problem. First pad both ends of data and convolution result with zeros so that the convolution emulates polynomial multiplication. Then do a division to get the kernel as a polynomial.

data1 = {0., 0., 1., 2., 1., 2., 1., 0., 0., 0.};
ker = {1/3, 1/3, 1/3};
conv = ListConvolve[ker, data1, 2];

Augment:

data2 = Join[{0.}, data1, {0.}];
conv2 = Join[{0., 0.}, conv, {0., 0.}];

Rewrite as explicit polynomials:

polyfactor = data2.x^Range[0, Length[data2] - 1];
polyprod = conv2.x^Range[0, Length[conv2] - 1];

Compute the kernel as coefficients of the polynomial quotient:

CoefficientList[Chop[PolynomialQuotient[polyprod, polyfactor, x]], x]

(* {0.333333333333, 0.333333333333, 0.333333333333} *)

If actual examples are large and efficiency is a concern, there are equivalent but faster ways to recover the kernel e.g. emulating the division using a Fourier transform.

To approximate ker using ListDeconvolve, use data1 as the kernel. Increasing MaxIterations can improve the approximation.

data1 = {0., 0., 1., 2., 1., 2., 1., 0., 0., 0.};

ker = {1/3, 1/3, 1/3};

conv = ListConvolve[ker, data1, 2];

Column[ListDeconvolve[data1, conv, MaxIterations -> #, 
     Method -> "RichardsonLucy"][[5 ;; 7]] & /@ Range[10, 40, 10]]