Let $\alpha=\sum_i a_i$, $[\alpha]=\{1,2,\ldots,\alpha\}$, and $\mathcal S=\{A,B,C,D,E,F\}$. Define for each $j\in[\alpha]$
$$z_j = \begin{cases}1,& \text{ team $j$ is used}\\0,& \text{ otherwise}, \end{cases} $$
and define for $(i,j)\in \mathcal S\times[\alpha]$ $$x_{ij}=\begin{cases}1,& \text{ site $i$ is visited by team $j$ }\\0,& \text{ otherwise}. \end{cases} $$
We can model this problem as the mixed-integer linear program:
\begin{align}
\min&\quad \sum_{j=1}^\alpha z_j\\
\mathrm{s.t.}&\quad \sum_{j=1}^\alpha x_{ij}\geqslant a_i, \quad\quad\quad i\in\mathcal S \tag1\\
&\quad x_{ij} \leqslant z_j, \quad\quad\quad\quad\;\, (i,j)\in\mathcal S\times[\alpha]\tag2\\
&\quad \sum_{i\in\mathcal S}x_{ij} \geqslant 3z_j, \quad\quad\; (i,j)\in\mathcal S\times[\alpha]\tag3\\
&\quad \sum_{i\in\mathcal S}x_{ij} \leqslant 4z_j, \quad\quad\quad\quad (i,j)\in\mathcal S\times[\alpha]\tag4\\
&\quad x_{Cj} + x_{Dj} \leqslant 1, \quad\quad j\in[\alpha] \tag5\\
&\quad x_{Cj} + x_{Ej} \leqslant 1, \quad\quad j\in[\alpha]\tag6\\
&\quad x_{Aj} + x_{Bj} \leqslant 1, \quad\quad j\in[\alpha]\tag7\\
&\quad x_{Bj} + x_{Fj} \leqslant 1, \quad\quad j\in[\alpha]\tag8\\
&\quad x_{Dj} + x_{Fj} \leqslant 1, \quad\quad j\in[\alpha]\tag9\\
&\quad z_j\in\{0,1\},\quad\quad\quad\;\, j\in[\alpha]\\
&\quad x_{ij}\in\{0,1\},\quad\quad\quad i,j\in\mathcal S\times[\alpha]
\end{align}
Constraints $(1)$ satisfy demand. Constraints $(2)$ require that team $j$ be used in order for any site to be visited by team $j$. Constraints $(3)$ and $(4)$ enforce the restriction on the number of sites that a given team visits. Constraints $(5)$-$(9)$ enforce the conflicts of interest.
Note: I chose $\sum_i a_i$ as an upper bound for the number of teams because it works, but a better upper bound would likely result in a formulation that is much easier to solve.
