#JavaScript (ES6), 75 bytes

Saved 2 bytes thanks to @KevinCruijssen

Expects a 0-based index.

(h,w,i)=>[...'12221000'].map((k,j,a)=>(i+w+~-k)%w+~~(i/w+h+~-a[j+2&7])%h*w)

Try it online!

The surrounding indices are returned in the following order:

$$\begin{matrix}5&4&3\\6&\cdot&2\\7&0&1\end{matrix}$$

###How?

The indices \$I_{dx,dy}\$ of each surrounding cell at \$(x+dx,y+dy)\$ are given by:

$$\begin{align}I_{dx,dy}&=((x+dx) \bmod w)+w((y+dy) \bmod h)\\&=((N+dx) \bmod w)+w((\left\lfloor\frac{N}{w}\right\rfloor+dy) \bmod h)\end{align}$$

where \$N=wy+x\$ is the index of the target cell.

We walk through the list \$[1,2,2,2,1,0,0,0]\$ and subtract \$1\$ to get the value of \$dx\$, which gives:

$$[0,1,1,1,0,-1,-1,-1]$$

For the corresponding values of \$dy\$, we use the same list shifted by 2 positions, which gives:

$$[1,1,0,-1,-1,-1,0,1]$$

JavaScript (ES6), 75 bytes

Saved 2 bytes thanks to @KevinCruijssen

Expects a 0-based index.

(h,w,i)=>[...'12221000'].map((k,j,a)=>(i+w+~-k)%w+~~(i/w+h+~-a[j+2&7])%h*w)

Try it online!

The surrounding indices are returned in the following order:

$$\begin{matrix}5&4&3\\6&\cdot&2\\7&0&1\end{matrix}$$

How?

The indices \$I_{dx,dy}\$ of each surrounding cell at \$(x+dx,y+dy)\$ are given by:

$$\begin{align}I_{dx,dy}&=((x+dx) \bmod w)+w((y+dy) \bmod h)\\&=((N+dx) \bmod w)+w((\left\lfloor\frac{N}{w}\right\rfloor+dy) \bmod h)\end{align}$$

where \$N=wy+x\$ is the index of the target cell.

We walk through the list \$[1,2,2,2,1,0,0,0]\$ and subtract \$1\$ to get the value of \$dx\$, which gives:

$$[0,1,1,1,0,-1,-1,-1]$$

For the corresponding values of \$dy\$, we use the same list shifted by 2 positions, which gives:

$$[1,1,0,-1,-1,-1,0,1]$$

#JavaScript (ES6), 77 bytes

Expects a 0-based index.

(h,w,i)=>[...'12221000'].map((k,j,a)=>(i+w+~-k)%w+w*(~~(i/w+h+~-a[j+2&7])%h))

Try it online!

The surrounding indices are returned in the following order:

$$\begin{matrix}5&4&3\\6&\cdot&2\\7&0&1\end{matrix}$$

###How?

The indices \$I_{dx,dy}\$ of each surrounding cell at \$(x+dx,y+dy)\$ are given by:

$$\begin{align}I_{dx,dy}&=((x+dx) \bmod w)+w((y+dy) \bmod h)\\&=((N+dx) \bmod w)+w((\left\lfloor\frac{N}{w}\right\rfloor+dy) \bmod h)\end{align}$$

where \$N=wy+x\$ is the index of the target cell.

We walk through the list \$[1,2,2,2,1,0,0,0]\$ and subtract \$1\$ to get the value of \$dx\$, which gives:

$$[0,1,1,1,0,-1,-1,-1]$$

For the corresponding values of \$dy\$, we use the same list shifted by 2 positions, which gives:

$$[1,1,0,-1,-1,-1,0,1]$$

#JavaScript (ES6), 75 bytes

Saved 2 bytes thanks to @KevinCruijssen

Expects a 0-based index.

(h,w,i)=>[...'12221000'].map((k,j,a)=>(i+w+~-k)%w+~~(i/w+h+~-a[j+2&7])%h*w)

Try it online!

The surrounding indices are returned in the following order:

$$\begin{matrix}5&4&3\\6&\cdot&2\\7&0&1\end{matrix}$$

###How?

The indices \$I_{dx,dy}\$ of each surrounding cell at \$(x+dx,y+dy)\$ are given by:

$$\begin{align}I_{dx,dy}&=((x+dx) \bmod w)+w((y+dy) \bmod h)\\&=((N+dx) \bmod w)+w((\left\lfloor\frac{N}{w}\right\rfloor+dy) \bmod h)\end{align}$$

where \$N=wy+x\$ is the index of the target cell.

We walk through the list \$[1,2,2,2,1,0,0,0]\$ and subtract \$1\$ to get the value of \$dx\$, which gives:

$$[0,1,1,1,0,-1,-1,-1]$$

For the corresponding values of \$dy\$, we use the same list shifted by 2 positions, which gives:

$$[1,1,0,-1,-1,-1,0,1]$$

#JavaScript (ES6), 77 bytes

Expects a 0-based index.

(h,w,i)=>[...'12221000'].map((k,j,a)=>(i+w+~-k)%w+w*(~~(i/w+h+~-a[j+2&7])%h))

Try it online!

The surrounding indices are returned in the following order:

$$\begin{matrix}5&4&3\\6&\cdot&2\\7&0&1\end{matrix}$$

###How?

The indices \$I_{dx,dy}\$ of each surrounding cell at \$(x+dx,y+dy)\$ are given by:

$$\begin{align}I_{dx,dy}&=((x+dx) \bmod w)+w((y+dy) \bmod h)\\&=((N+dx) \bmod w)+w((\left\lfloor\frac{N}{w}\right\rfloor+dy) \bmod h)\end{align}$$

where \$N=wy+x\$ is the index of the target cell.

We walk though the list \$[1,2,2,2,1,0,0,0]\$ and subtract \$1\$ to get the value of \$dx\$, which gives:

$$[0,1,1,1,0,-1,-1,-1]$$

For the corresponding values of \$dy\$, we use the same list shifted by 2 positions, which gives:

$$[1,1,0,-1,-1,-1,0,1]$$

#JavaScript (ES6), 77 bytes

Expects a 0-based index.

(h,w,i)=>[...'12221000'].map((k,j,a)=>(i+w+~-k)%w+w*(~~(i/w+h+~-a[j+2&7])%h))

Try it online!

The surrounding indices are returned in the following order:

$$\begin{matrix}5&4&3\\6&\cdot&2\\7&0&1\end{matrix}$$

###How?

The indices \$I_{dx,dy}\$ of each surrounding cell at \$(x+dx,y+dy)\$ are given by:

$$\begin{align}I_{dx,dy}&=((x+dx) \bmod w)+w((y+dy) \bmod h)\\&=((N+dx) \bmod w)+w((\left\lfloor\frac{N}{w}\right\rfloor+dy) \bmod h)\end{align}$$

where \$N=wy+x\$ is the index of the target cell.

We walk through the list \$[1,2,2,2,1,0,0,0]\$ and subtract \$1\$ to get the value of \$dx\$, which gives:

$$[0,1,1,1,0,-1,-1,-1]$$

For the corresponding values of \$dy\$, we use the same list shifted by 2 positions, which gives:

$$[1,1,0,-1,-1,-1,0,1]$$