  We consider simple supercuspidal representations of $\mathrm{GSp}_4$ over a
$p$-adic field and show that they have conductor exponent 5. We study
(paramodular) newvectors and minimal vectors in these representations, obtain
formulas for their matrix coefficients, and compute key local integrals
involving these as test vectors.
  Our local computations lead to several explicit global period formulas
involving automorphic representations $\pi$ of $\mathrm{GSp}_4(\mathbb{A})$
whose local components (at ramified primes) are simple supercuspidal
representations, and where the global test vectors are chosen to be (diagonal
shifts of) newforms or automorphic forms of minimal type. As an analytic
application of our work to the sup-norm problem, we show the existence of
paramodular newforms on $\mathrm{GSp}_4(\mathbb{A})$ of conductor $p^5$ that
take ``large values" on a fixed compact set as $p\rightarrow \infty$.
