Yes. Indeed, for $X$ a set, let $G_X$ be the group of partial bijections of $X$, that are defined and identity outside a countable subset. I claim that, for $X$ uncountable, every countable subset of $G$ is contained in a (5-generator) finitely generated submonoid (and hence in a finitely generated inverse submonoid).

The claim being granted, and using that the power set of $\omega$ contains a chain isomorphic to $(\mathbf{Q},\le)$, one obtains such a chain of idempotents in a suitable inverse monoid.

Note: the same claim was proved by Sierpinski and Banach in the 1930's for the monoid of all self-maps of every set, and by Galvin (1995) for the group of all permutations of every set.

Now let me prove the claim, inspired by Galvin's proof. Let $(f_n)_{n\in\mathbf{Z}}$ be a sequence in $G_X$. So there exists an infinite countable subset $X_{0,0}$ such that for every $n$, each $f_n$ is defined and identity outside $X_{0,0}$. Choose for all other $(m,n)\in\mathbf{Z}^2$ an infinite countable susbet $X_{m,n}$, pairwise disjoint. Henceforth, all maps are assumed to be defined and identity outside $X'=\bigcup_{m,n}X_{m,n}$. Also fix a bijection $X_{0,0}\to X_{m,n}$ for all $(m,n)\neq (0,0)$, so that we identify $X'$ to $X_{0,0}\times\mathbf{Z}^2$.

Define

$u$ as the permutation $(x,m,n)\mapsto (x,m+1,n)$;

$r$ as the permutation $(x,0,n)\mapsto (x,0,n+1)$, $(x,m,n)\mapsto (x,m,n)$ for $m\neq 0$;

$f$ as the permutation $(x,m,n)\mapsto (f_m(x),m,n)$ for $n\ge 0$ and $(x,m,n)\mapsto (x,m,n)$ for $n\ge 0$.

I claim that for every $m$ we have $f_m\in\langle u,u^{-1},r,r^{-1},f\rangle$, where $\langle\cdots\rangle$ means the submonoid generated (actually, it follows that $f_m\in\langle u,r,f\rangle_{\text{inverse-monoid}}$).

Indeed, write $g_m=u^mfu^{-m}$: then $g_m$ is like $f$, but shifted $m$ times to the right. Then one sees that $g_m(r^{-1}g_mr)^{-1}=f_m$, and the claim is proved.

[Note 1: observe that $f_m$ is written as a word of length $\le 2+2(2m+1)=4m+6$ with respect to the given generators: since this only depends on $m$, this shows that $G_X$ is "strongly distorted" (as monoid, and as inverse monoid) and in particular strongly bounded, a.k.a. Bergman's property.]

[Note 2: Probably it's also true for $X$ countable, with some further preliminary lemmas. Also with only two generators.]

[Note 3: from Vagner-Preston, every countable inverse monoid embeds into $G_{\aleph_1}$. As corollary, every countable inverse monoid embeds into a 3-generated one. This is probably well-known?]