I am thinking about infinite loop spaces and spectra. The category of connective spectra is in fact equivalent to the category of infinite loop spaces.
Is it also the case that the category of spectra can be formed from sequences of infinite loop spaces? Instead of sequential spectra, this would entail sequential infinite loop spaces.
You are essentially asking if the category of (sequential) spectra is equivalent to the category of sequential spectrum objects in the category of infinite loop spaces. Now, since this is a homotopy theoretical question in all ways, I hope you can agree that it is more interesting (and model-independent) to ask if the $\infty$-category of spectra $\mathsf{Sp}$ is equivalent to the $\infty$-category $\mathsf{Sp}(\mathsf{Sp}_\mathrm{conn})$ of spectrum objects in the $\infty$-category of connective spectra. (By definition, $\mathsf{Sp}\simeq\mathsf{Sp}(\mathsf{Spaces})$, where $\mathsf{Spaces}$ is the $\infty$-category of spaces.) We use that connective spectra/infinite loop spaces are equivalent to $\mathbb{E}_\infty$-groups, i.e. homotopy coherent versions of commutative groups in the $\infty$-category of spaces. For a general $\infty$-category $\mathscr{C}$ with finite limits, we denote the $\infty$-category of homotopy coherent commutative group objects in $\mathscr{C}$ by $\mathrm{CGrp}(\mathscr{C})$. One can show that, for a general $\infty$-category $\mathscr{C}$ with finite limits, the forgetful functor $\mathsf{Sp}(\mathrm{CGrp}(\mathscr{C}))\to\mathsf{Sp}(\mathscr{C})$ is an equivalence, so in particular, the forgetful functor $\mathsf{Sp}(\mathsf{Sp}_\mathrm{conn})\to\mathsf{Sp}$ is an equivalence of $\infty$-categories.
This general statement and its proof are Proposition II.23 in lecture notes written by Ferdinand Wagner of Fabian Hebestreit's course on algebraic and hermitian $K$-theory. The idea of the proof is that there is an equivalence $\mathsf{Sp}(\mathrm{CGrp}(\mathscr{C}))\simeq\mathrm{CGrp}(\mathsf{Sp}(\mathscr{C}))$ by purely formal reasons, essentially based on facts that limits commute with limits and the like. Then we use that in an additive $\infty$-category $\mathscr{D}$, the forgetful functor gives an equivalence $\mathrm{CGrp}(\mathscr{D})\to\mathscr{D}$ (just like in $1$-category land), and since $\mathsf{Sp}(\mathscr{C})$ is indeed an additive $\infty$-category (it is even a stable $\infty$-category, but that hasn't been shown yet at this point in the lecture notes), we are done.
The statement that $\mathsf{Sp}_\mathrm{conn}\simeq\mathrm{CGrp}(\mathsf{Spaces})$ is Corollary II.24 in the same lecture notes.