Here are some inconclusive thoughts.
To construct (reasonable) limits of locally presentable categories, it suffices to consider products, inserters, and equifiers.
It's easy to check that locally $\kappa$-presentable categories are closed under small products. For colimits can be computed pointwise, and a generating set is given by objects whose components are $\kappa$-presentable, and initial except on a $\kappa$-small set of coordinates. If we were working with general $\kappa$-accessible categories, this would be more complicated -- I'm not sure, we might have to raise the degree of accessibility.
I suspect that when $\kappa$ is uncountable, locally $\kappa$-presentable categories are closed under inserters $Ins(F,G)$ between functors $F,G: \mathcal{C} \to \mathcal{D}$ which are cocontinuous and preserve the $\kappa$-presentable objects (equivalently, functors with a $\kappa$-accessible right adjoint). I don't have a complete proof. Colimits are computed as in $\mathcal{C}$, and it's easy to see that the objects whose underlying $\mathcal{C}$-object is $\kappa$-presentable are $\kappa$-presentable. It suffices to show that the colimit closure of these objects is all of $Ins(F,G)$ -- by Theorem 2.5.1 in Makkai-Paré, or by Lemma 3.7 in these notes by Mike Shulman and the usual strong-generator characterization of locally $\kappa$-presentable categories. If $F(c) \overset{\gamma}{\to} G(c)$ is an object of $Ins(F,G)$, I would like to induct on the presentability rank $\lambda$ of $c$ to show that $F(c) \overset{\gamma}{\to} G(c)$ has the same presentability rank $\lambda$. The idea is that $c$ is a $\lambda$-small colimit of $\kappa$-presentable objects (see Adámek and Rosický Remark 1.30). If $\lambda = \mu^+$, then this means that $c$ is a $\mu$-sized colimit of $\kappa$-presentable objects. By adding $\mu$-small colimits to the indexing diagram, $c$ is in fact a $\mu$-sized, $\mu$-filtered colimit of $\mu$-presentable objects, and there must be a cofinal chain. So $c$ is the colimit of a $\mu$-sized chain of $\mu$-presentable objects $c = \varinjlim m_i$, which by induction are in the colimit closure we're interested in; moreover we can assume that the diagram functor preserves colimits of chains. Now by $\mu$-presentability, for every $m_i$ the map $\gamma$ factors through $F m_i \to G m_{u(i)}$; we can assume $u(i) \geq i$, that $u$ is order-preserving [this is the reason we passed to a chain, so that order-preservation is almost free], and that $u$ preserves colimits of chains. Look at the sequence $i, u(i), u(u(i)), \dots$. The $\omega$th iterate must be a fixed point [this is where we use that $\kappa$ is uncountable], so the fixed points form a cofinal set $J \subseteq I$. Then we can form $\varinjlim_{j \in J} (F m_j \to G m_j)$ as a colimit in $Ins(F,G)$ to complete the induction.
The problem is that this induction only works when $\lambda$ is the the successor of a regular cardinal $\mu$ (I think it's fine if $\lambda$ is a regular limit cardinal, but it breaks down if $\lambda$ is the successor of a singular cardinal $\mu$ because the $\mu$-presentable objects don't actually have lower presentability rank than $c$ so the induction gets stuck -- the first issue is when $\lambda = \aleph_{\omega + 1}$), so the argument is not complete. But the fact that the only issues come from singular cardinals suggests to me that the issues are purely an artifact of using too naive of a method.
I haven't thought too much about equifiers, but I suspect that similar considerations are relevant as in the inserter case.