An empty regiment has no least glorious member, and I wouldn't put it past Arstotzka to make empty regiments. For nonempty regiments, though, and counting two soldiers as both least glorious if they tie...
For there to be no least glorious soldier in a nonempty regiment, we need an infinite sequence of less glorious soldiers. Assume for contradiction we have such a sequence $(S_n)_{n \in \mathbb{N}}$.
Number the medal types from 1 to 7, and denote the number of medals of type $k$ a soldier $S$ has as $m_k(S)$. The sequence $(m_1(S_n))_{n \in \mathbb{N}}$ of type-1 medal counts of the $S_n$ soldiers is an infinite sequence of nonnegative integers, and thus must contain an infinite non-decreasing subsequence. Let $(S_n^1)_{n \in \mathbb{N}}$ be a subsequence of $(S_n)_{n \in \mathbb{N}}$ with non-decreasing numbers of type-1 medals.
Similarly, $(S_n^1)_{n \in \mathbb{N}}$ must contain an infinite subsequence $(S_n^2)_{n \in \mathbb{N}}$ with non-decreasing numbers of type-2 medals (and non-decreasing numbers of type-1 medals, as it's a subsequence of $(S_n^1)_{n \in \mathbb{N}}$), and $(S_n^2)_{n \in \mathbb{N}}$ must contain an infinite subsequence $(S_n^3)_{n \in \mathbb{N}}$ with non-decreasing type-3 medals, and so on all the way up to $(S_n^7)_{n \in \mathbb{N}}$, which has non-decreasing numbers of every type of medal.
However, for one soldier to be less glorious than another, the first soldier must have less of at least one medal type than the other. $(S_n^7)_{n \in \mathbb{N}}$ was constructed as a subsequence of a sequence of successively less glorious soldiers, but its soldiers cannot be successively less glorious.
This is a contradiction, so the original regiment cannot have an infinite sequence of less glorious soldiers, and it must thus have a least glorious soldier.