I know how to find the the order of vanishing for a complex $L$-series $L(E,1)$.
I'm looking at an example:
$\mathcal{L}_7(E,T)=O(7^9)+O(7^6)T+(6\cdot7^5+O(7^6))T^2+(3\cdot7^5+O(7^6))T^3+(5+5\cdot7+2\cdot7^4+7^5+O(7^6))T^4+O(T^5)$
How do I tell that the $L$-series vanishes to order 7?
Firstly, you have to figure out what $T$ means. Often (indeed, typically) for $p$-adic $L$-functions, $T$ denotes an element in the Iwasawa algebra, chosen so that the Iwasawa algebra equals $\mathbb Z_p[[T]]$, and then there is a change of variables to relate this to the classical $s$ variable, e.g. of the form $T = (1+p)^{s-1} - 1.$
But different people use different conventions, and you will have to figure out what conventions your example is using.
Let's suppose, though, that it is the one I gave, so that $s = 1$ corresponds to $T = 0$. Then your example certainly doesn't vanish to order $7$. Indeed, it has a non-zero coefficient of $T^2$, and may possibly have a non-zero constant term or non-zero coefficient of $T$ as well (the information you give isn't dispositive on this).
So the order of vanishing is at most $2$.
On the other hand, if the convention for relating $T$ and $s$ is different, e.g. if $T = (1+p)^{s} - 1,$ then you would have to compute the order of vanishing at $T = 7$, and this would be different. (Although since you don't have terms all the way up to degree $7$, even if the order of vanishing was $7$, I don't think you would be able to see that with the degree of approximation you have.)
One more thing: it's pretty hard to find elliptic curves of rank $7$, so I would be surprised if you really had an $L$-function that had that much vanishing at $s = 1$. But they do exist, so maybe you've found one!
Oh, and for background, the paper of Mazur and Swinnerton-Dyer is still pretty good.