Is CBC mode encryption vulnerable to a reordering attack?
Yes.  Assume that the attacker knows the ciphertext $c = c_1 \mathbin\| c_2$, the initialization vector $v$ and the plaintext $m = m_1 \mathbin\| m_2$.  This tells them that


$D_k(c_1) = m_1 \oplus v$ and
$D_k(c_2) = m_2 \oplus c_1$,


where $D_k(\cdot)$ denotes block cipher decryption under the (unknown) key $k$.

In particular, this implies that, if the attacker reverses the order of the ciphertext blocks to produce the modified ciphertext $c' = c_2 \mathbin\| c_1$, then they will know that it will decrypt to $m' = m'_1 \mathbin\| m'_2$, where


$m'_1 = D_k(c_2) \oplus v = m_2 \oplus c_1 \oplus v$ and
$m'_2 = D_k(c_1) \oplus c_2 = m_1 \oplus c_2 \oplus v$.


There should be nothing particularly surprising about this: like all non-authenticated block cipher modes of operation, CBC mode is known to be malleable, meaning that an attacker who can change the ciphertext is also able, at least to some extent, to predict the effects of the changes on the resulting plaintext.  If you don't want that (and you usually don't), you need to use an authenticated encryption mode, possibly constructed by combining a conventional mode like CBC with a secure MAC.

(In fact, a further attack on CBC is possible if, as usual, the initialization vector $v$ is transmitted alongside the ciphertext: in that case, the attacker can change the first block of the plaintext to anything they want, without affecting the other blocks, simply by changing the IV.)