There's a huge list of possibilities. The easiest place to draw from is historical effects of wars in human history. For starters, here's one about the US Civil War. Any good historical source for a war will give you things to work with, though.
Let's look at a few specific issues that may crop up:
During wartime, especially with a paranoid leadership, travel restrictions are normal. Fear of the "other" grows. If the party doesn't look like they belong in the empire in question, they will get harassed by soldiers and shunned by civilians. If they have official paperwork (or good forgeries), they will be able to get past the soldiers at least.
If they look like they fit in, or have a reputation for heroic acts for the empire, they won't have the same problems.
On top of official travel restrictions, movement in areas affected by the war is just plain more difficult. Ground is chewed up and turned into mud by army camps, bridges may be destroyed to stop an advancing force, soldiers may be up in trees shooting anybody not on their side, and so on.
Internment & Deportations
If one of them happens to have some Goblin in his lineage, he's facing a risk of being deported entirely, or thrown into an internment camp. This happens indiscriminately based on race.
Refugees & Empty Land
Since your empire is surrounded by Goblins and at war, it's likely there are fights around the border, and raids into nearby territory. That territory is going to be abandoned by most civilians. Farmland will lie fallow (and be trampled by armies), food will be sized to feed those armies, and the land will generally be stripped bare. Anybody staying in the area runs the risk of being collateral damage. If the civilians happen to be encountered by soldiers of the other side, war crimes are entirely possible.
That means for the PCs working in that territory, supplies will be harder to scavenge from the land. There won't be anybody selling anything, and there won't be easy game hunting. They could be attacked at any time, or confronted by a hungry army and have their things confiscated.
Deeper in the empire, there will be refugees that have to be dealt with somehow. In the best case scenario, they will be taken in and cared for. In a worse scenario, they'll be put in camps on the outskirts of existing settlements and have to fend for themselves. If not properly managed, crime can get very bad in these places.
Both armies and refugee camps often have disease problems, due to so many people being close together and the difficulty of maintaining proper hygiene in war conditions (and the general lack of sanitation in many poorly run refugee camps).
If the PCs stay away from both of those things this will impact them less than it will impact the general population, but a population afraid of an epidemic isn't going to be as welcoming to random people coming into town like the PCs. If they deal with soldiers or refugees at all, the disease risk is very real.
Conscription & Demographics
Is the empire low on soldiers? If so, the PCs may simply get conscripted into the army, under penalty of execution.
If this is a large war that has been going a while, there are demographic impacts to that. The bulk of a big army is young men. That has impacts at home, as you have a shortage of young men. That impacts everything from trying to start a family (there are a lot more women then men) to gender roles in society (female labour became more acceptable in the West because of a lack of men to do the jobs). Children may also be required to start work below the age of what is commonly regarded as acceptable if labour shortages become critical.
Good news if the PCs happen to be looking for wives. Bad news if they're looking for husbands, or to hire male workers for pretty much any purpose.
Equipment Price Inflation
During major wars, resources are redirected towards military purposes. While some things will be still available, prices go up and some things simply can't be legally obtained at any price. If the PCs need to get armor made and all the metal has been provisioned for military use... they either have to go to the black market or are just out of luck.
This can cover a huge variety of supplies, including metal, leather, food, wood, and pretty much anything else that's not unlimited and has a military use. Expertise is also limited, as people capable of making weapons & armor are going to be pulled in for military production first. The bigger scale the war is, the more this will be a problem for the PCs.(A minor border conflict won't be a big deal, but they will find it extremely difficult to operate if the empire is bending it's primary resources on the war).
This is not an exhaustive list, because so many wars have caused so many different consequences that listing them all here is impossible. What you'll want to do is figure out how major the war is, and then apply various problems for the PCs appropriately to that.
The really important thing is that it can't just be business as usual. They can't go around doing whatever they want as if no war is going on at all, because everybody else around them is being affected by it. Even if they don't want to get involved in it personally, it's going to come to them in some way.