Just to record an answer: yes, it is believed to be true that for each prime $p$ there is an elliptic curve $E_{\mathbb{Q}}$ such that $\underline{III}(\mathbb{Q},E)$ contains an element of order $p$. However, this is wide open.
One can however prove the following:
Theorem (R. Kloosterman): If you fix a prime $p$, then there is a number field $K_p$ -- i.e., depending on $p$ -- and an elliptic curve $E/K_p$ whose Shafarevich-Tate group contains an element of order $p$.
As Kloosterman remarked (in his 2005 paper on the subject, and also just now in the above comments) via a simple "Weil restriction" argument, this implies that there exists an abelian variety $A_{/\mathbb{Q}}$ whose Shafarevich-Tate group contains an element of order $p$. But now the dimension of $A$ goes to infinity with $p$.
Others have since worked out various improvements of Kloosterman's Theorem. Here are three different directions:
What is the minimal degree $d_p = [K_p:\mathbb{Q}]$ of a number field $K_p$ in above theorem? Kloosterman's argument gave $d_p = O(p^4)$. In a 2005 paper I showed that one can take $d_p = 2p^3$. In a 2010 paper with Shahed Sharif, we showed that one can take $d_p = p$. (Note that whether $d_p = p$ is in fact best possible is an open question. At one point I thought I had an argument to show that it was, but this was wrong. I now suspect that this is only best possible by a method proceeding along the lines of our construction. For instance, I believe it is conceivable that $d_p = 2$ for all $p$, and this has something to do with how Sha behaves in quadratic twists...)
What kind of elliptic curves $E$ can be used to produce these elements in Sha? Sharif and I showed that in fact one can start with any elliptic curve $E_0$ over $\mathbb{Q}$ and take its base change to $K_p$ a degree $p$ number field. (Similarly, one can start with any elliptic curve $E$ over any number field and get order $p$ elements of Sha in an extension field of degree $p$.) And in fact $p$ does not need to be a prime number here: it holds for every integer $n$. And in fact you can get as many elements of order $n > 1$ as you want.
What kind of control can one get over the field extension $K_p/\mathbb{Q}$? Can one for instance choose it to be Galois, abelian, or cyclic? Matsuno proves that for any cyclic degree $p$ extension $K_p/\mathbb{Q}$ one can find elliptic curves $E_{\mathbb{Q}}$ such that the base change to $K_p$ has as many order $p$ elements in its Shafarevich-Tate group as one wants. There are similar results along these lines (with dihedral extensions and elements of the Selmer group) by Alex Bartel.