Can the Walsh Hadamard transform be computed for non power of 2 sizes?
The Walsh-Hadamard transform requires a Hadamard matrix -- which has entries $\pm 1$ and
whose rows are orthogonal vectors.  A $n \times n$ Hadamard matrix is known to exist
for $n = 2$. Larger Hadamard matrices can exist only if $n$ is a multiple of $4$, though
it is not known if there is a Hadamard matrix for every multiple of $4$. However,
there is a recursive construction that can be used to construct a $2^m \times 2^m$
Hadamard matrix from a $2^{m-1}\times 2^{m-1}$ Hadamard matrix, and this structure 
allows for the use of the Fast Hadamard Transform algorithm, which reduces the
computational cost from $2^{2m}$ additions and subtractions to $m2^m$ additions and
subtractions (just like the $N$-FFT reduces the number of multiplications from
$O(N^2)$ to $O(N\log N)$.

With this as background, the answer is Yes, you can use a Walsh-Hadamard transform
of length $N$ if you can find a $N\times N$ Hadamard matrix, but your choices
for $N$ are necessarily restricted. Also, fast algorithms may not exist
for your choice of $N$, though some speed-up is usually possible.

Note that Walsh-Hadamard transforms (WHTs) do not support cyclic convolutions, but
they do support what is sometimes called Poisson convolution. If $H$ denotes 
a $2^m\times 2^m$
Hadamard matrix and $\mathbf x$ and $\mathbf y$ are vectors of length $2^m$ with \WHTs $\mathbf xH$ and $\mathbf yH$,
then the inverse WHT of the term-by-term multiplication of the entries
in $\mathbf xH$ and $\mathbf yH$ can be described as follows.  Suppose that the entries in
$\mathbf x$ and $\mathbf y$ etc are indexed not by integers $0$ through $2^m-1$ but rather
by the $m$-bit representations of these numbers.  Thus, we talk not of
$x[k]$ as the $k$-th entry in $\mathbf x$ but rather of $x[\mathbf k]$ where
$\mathbf k$ is the $m$-bit representation of $k$.  Then, the iWHT of the term-by-term
product of the entries in $\mathbf xH$ and $\mathbf yH$ has $\mathbf k$-th entry
$$\sum_{\mathbf i} x[\mathbf i]y[\mathbf k\oplus \mathbf i]$$
which is eerily reminiscent of
$$\sum_i x[i]y[k-i]$$
and even more so if one notes that modulo two, addition and subtraction
are the same and so that $\mathbf k\oplus \mathbf i$ could as well
have been written as $\mathbf k\ominus \mathbf i$.