Should I take MOF (metal-organic framework) as a periodic structure or an isolated cluster? What's the difference between them? If it's periodic, can I do an analysis of the electronic structure, such as band structure and density of states? If it's an isolated cluster, do I need to put a large vacuum along with the $x$, $y$, and $z$ directions? In essence, is there any difference between isolated clusters and periodic solid when modeling MOF structures?
There is no article better suited for your question than "Electronic Structure Modeling of Metal–Organic Frameworks" by Mancuso and coworkers. There is also some nice discussion about periodic structures versus cluster models in "Computational Design of Functionalized Metal–Organic Framework Nodes for Catalysis".
Whether you choose to model a MOF as a periodic structure or isolated cluster is a matter of preference and depends on the property you wish to model. The only way to get reasonable band structures and density of states is to use a periodic structure of a MOF (i.e. the full crystalline unit cell). If you are using a plane-wave density functional theory (DFT) code like VASP, as alluded to in your other questions, I also generally do not recommend using a cluster model. In principle, you can use a cluster model in a periodic DFT code like VASP with artificial vacuum space surrounding the cluster to avoid self-interactions, but there is rarely a justifiable reason to do this. DFT codes that use Gaussian basis sets and meant for molecular systems are far better suited for using cluster models.
This begs the question – why would you ever use a cluster model? Well, if you are studying a phenomenon that is relatively local (e.g. a catalytic reaction happening at a specific active site), it is unlikely that the framework atoms far away from the active site will influence the reactivity. In this case, it might make sense to carve a representative cluster of your MOF, ensuring that all dangling bonds are capped and the charge of your system is balanced. Using a cluster model also lets you model an inherently smaller system, which means you can afford to use more computationally expensive (yet hopefully more accurate) methods like (meta-)hybrid functionals or wavefunction theory. These would typically be impractical to do with a periodic structure.
Of course, using a cluster model has its disadvantages. By truncating the system, you may ignore subtle but important structural or electronic effects from the atoms you have essentially removed from consideration. For this reason, it is a good idea to do some benchmarking to ensure the properties of interested computed from the cluster model are largely representative of the original periodic system. Sometimes, in a given project it can be beneficial to use both approaches depending on what you wish to answer. There are also other approaches, such as QM/MM, that allow you to model a periodic structure at a computational cost approaching that of a typical cluster model.